Swap the domain and range values to get the inverse function. They are the inverse functions of the double exponential function, tetration, of f(w) = wew,[105] and of the logistic function, respectively.[106]. + That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. Therefore, the complex logarithms of z, which are all those complex values ak for which the ak-th power of e equals z, are the infinitely many values, Taking k such that Want some good news, free of charge? ≤ This is the currently selected item. This example graphs the common log: f(x) = log x. which is read “ y equals the log of x, base b” or “ y equals the log, base b, of x.” In both forms, x > 0 and b > 0, b ≠ 1. The polar form encodes a non-zero complex number z by its absolute value, that is, the (positive, real) distance r to the origin, and an angle between the real (x) axis Re and the line passing through both the origin and z. So the Logarithmic Function can be "reversed" by the Exponential Function. Definition of logarithmic function : a function (such as y= logaxor y= ln x) that is the inverse of an exponential function (such as y= axor y= ex) so that the independent variable appears in a logarithm First Known Use of logarithmic function 1836, in the meaning defined above The parent function for any log is written f(x) = log b x. We can shift, stretch, compress, and reflect the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] without loss of shape.. Graphing a Horizontal Shift of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] cos [103] Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field. Start studying Parent Functions - Odd, Even, or Neither. The illustration at the right depicts Log(z), confining the arguments of z to the interval (-π, π]. The black point at z = 1 corresponds to absolute value zero and brighter, more saturated colors refer to bigger absolute values. {\displaystyle \sin } Example 2: Using y=log10(x), sketch the function 3log10(x+9)-8 using transformations and state the domain & range. . log b y = x means b x = y.. Graphs of logarithmic functions. This is the currently selected item. You'll often see items plotted on a "log scale". A logarithmic function is a function of the form . φ Logarithmic Functions The "basic" logarithmic function is the function, y = log b x, where x, b > 0 and b ≠ 1. When the base is greater than 1 (a growth), the graph increases, and when the base is less than 1 (a decay), the graph decreases. The exponential … Because f(x) and y represent the same thing mathematically, and because dealing with y is easier in this case, you can rewrite the equation as y = log x. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential. The hue of the color encodes the argument of Log(z).|alt=A density plot. Sal is given a graph of a logarithmic function with four possible formulas, and finds the appropriate one. n, is given by, This can be used to obtain Stirling's formula, an approximation of n! Shape of a logarithmic parent graph. {\displaystyle \cos } The base of the logarithm is b. Let us come to the names of those three parts with an example. This reflects the graph about the line y=x. [100] Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Practice: Graphs of logarithmic functions. The inverse of an exponential function is a logarithmic function. We give the basic properties and graphs of logarithm functions. Graphing logarithmic functions according to given equation. Source(s): https://shorte.im/bbGNP. This angle is called the argument of z. The resulting complex number is always z, as illustrated at the right for k = 1. Ask Question + 100. For example, g(x) = log 4 x corresponds to a different family of functions than h(x) = log 8 x. There are no restrictions on y. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. Aug 25, 2018 - This file contains ONE handout detailing the characteristics of the Logarithmic Parent Function. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.]]. Rewrite each exponential equation in its equivalent logarithmic form. Logarithmic one-forms df/f appear in complex analysis and algebraic geometry as differential forms with logarithmic poles. , The discrete logarithm is the integer n solving the equation, where x is an element of the group. (Remember that when no base is shown, the base is understood to be 10.) log10A = B In the above logarithmic function, 10is called asBase A is called as Argument B is called as Answer The natural logarithm of a positive, real number a may be defined as the area under the graph of the hyperbola with equation y = 1/x between x = 1 and x = a.This is the integral = ∫. Still have questions? Graph of f(x) = ln(x) The parent function for any log has a vertical asymptote at x = 0. − The graph of an log function (a parent function: one that isn’t shifted) has an asymptote of \(x=0\). is within the defined interval for the principal arguments, then ak is called the principal value of the logarithm, denoted Log(z), again with a capital L. The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. The following steps show you how to do just that when graphing f(x) = log3(x – 1) + 2: First, rewrite the equation as y = log3(x – 1) + 2. The domain of function f is the interval (0 , + ∞). Intercepts of Logarithmic Functions By examining the nature of the logarithmic graph, we have seen that the parent function will stay to the right of the x-axis, unless acted upon by a transformation. Logarithmic Parent Function. Dropping the range restrictions on the argument makes the relations "argument of z", and consequently the "logarithm of z", multi-valued functions. Learn vocabulary, terms, and more with flashcards, games, and other study tools. NOTE: Compare Figure 6 to the graph we saw in Graphs of Logarithmic and Exponential Functions, where we learned that the exponential curve is the reflection of the logarithmic function in the line y = x. φ Vertical asymptote. Range: All real numbers . The function f(x)=ln(9.2x) is a horizontal compression t of the parent function by a factor of 5/46 Domain: x > 0 . Get your answers by asking now. In general, the function y = log b x where b, x > 0 and b ≠ 1 is a continuous and one-to-one function. We begin with the parent function Because every logarithmic function of this form is the inverse of an exponential function with the form their graphs will be reflections of each other across the line To illustrate this, we can observe the relationship between the … She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. The logarithm of x to base b is denoted as logb(x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation. Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. sin To solve for y in this case, add 1 to both sides to get 3x – 2 + 1 = y. [97] These regions, where the argument of z is uniquely determined are called branches of the argument function. [110], Inverse of the exponential function, which maps products to sums, Derivation of the conversion factor between logarithms of arbitrary base. You can change any log into an exponential expression, so this step comes first. The next figure shows the graph of the logarithm. Remember that since the logarithmic function is the inverse of the exponential function, the domain of logarithmic function is the range of exponential function, and vice versa. Here is a set of practice problems to accompany the Logarithm Functions section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. However, most students still prefer to change the log function to an exponential one and then graph. where a is the vertical stretch or shrink, h is the horizontal shift, and v is the vertical shift. Trending Questions. y = b x.. An exponential function is the inverse of a logarithm function. Exponential functions. It is called the logarithmic function with base a. Graphs of logarithmic functions. Corresponding to every logarithm function with base b, we see that there is an exponential function with base b:. The range of f is given by the interval (- ∞ , + ∞). Did you notice that the asymptote for the log changed as well? Common Parent Functions Tutoring and Learning Centre, George Brown College 2014 ... Natural Logarithmic Function: f(x) = lnx . Then subtract 2 from both sides to get y – 2 = log3(x – 1). y = logax only under the following conditions: x = ay, a > 0, and a1. • The parent function, y = logb x, will always have an x-intercept of one, occurring at the ordered pair of (1,0). As we mentioned in the beginning of the section, transformations of logarithmic functions behave similar to those of other parent functions. for large n.[95], All the complex numbers a that solve the equation. We begin with the parent function y = log b (x). Because you’re now graphing an exponential function, you can plug and chug a few x values to find y values and get points. Evidence for Multiple Representations of Numerical Quantity", "The Effective Use of Benford's Law in Detecting Fraud in Accounting Data", "Elegant Chaos: Algebraically Simple Chaotic Flows", Khan Academy: Logarithms, free online micro lectures, https://en.wikipedia.org/w/index.php?title=Logarithm&oldid=1001831533, Articles needing additional references from October 2020, All articles needing additional references, Articles with Encyclopædia Britannica links, Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 January 2021, at 15:40. k [108] The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. In my head, this means one side is counting "number of digits" or "number of multiplications", not the value itself. Again, this helps show wildly varying events on a single scale (going from 1 to 10, not 1 to billions). {\displaystyle -\pi <\varphi \leq \pi } Pierce (1977) "A brief history of logarithm", International Organization for Standardization, "The Ultimate Guide to Logarithm — Theory & Applications", "Pseudo Division and Pseudo Multiplication Processes", "Practically fast multiple-precision evaluation of log(x)", Society for Industrial and Applied Mathematics, "The information capacity of the human motor system in controlling the amplitude of movement", "The Development of Numerical Estimation. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay. Logarithmic functions are the inverses of exponential functions. Find the inverse function by switching x and y. Change the log to an exponential. {\displaystyle 0\leq \varphi <2\pi .} Moreover, Lis(1) equals the Riemann zeta function ζ(s). This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there. [109], The polylogarithm is the function defined by, It is related to the natural logarithm by Li1(z) = −ln(1 − z). and their periodicity in So if you can find the graph of the parent function logb x, you can transform it. are called complex logarithms of z, when z is (considered as) a complex number. The parent function for any log is written f(x) = logb x. ... We'll have to raise it to the second power. Practice: Graphs of logarithmic functions. But it is more common to write it this way: f(x) = ln(x) "ln" meaning "log, natural" So when you see ln(x), just remember it is the logarithmic function with base e: log e (x). You now have a vertical asymptote at x = 1. 2 If y – 2 = log3(x – 1) is the logarithmic function, 3y – 2 = x – 1 is the exponential; the inverse function is 3x – 2 = y – 1 because x and y switch places in the inverse. Such a number can be visualized by a point in the complex plane, as shown at the right. {\displaystyle \varphi +2k\pi } {\displaystyle 2\pi ,} [104], Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. Select from the drop-down menus to correctly identify the parameter and the effect the parameter has on the parent function. Reflect every point on the inverse function graph over the line y = x. [107] By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx/x on the positive reals. The natural logarithm can be defined in several equivalent ways. π This is not the same situation as Figure 1 compared to Figure 6. From the perspective of group theory, the identity log(cd) = log(c) + log(d) expresses a group isomorphism between positive reals under multiplication and reals under addition. Because f(x) and y represent the same thing mathematically, and because dealing with y is easier in this case, you can rewrite the equation as y = log x. < < Switch every x and y value in each point to get the graph of the inverse function. The Natural Logarithm Function. Vertical asymptote of natural log. The next figure illustrates this last step, which yields the parent log’s graph. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.[99]. The graph of 10x = y gets really big, really fast. π Its horizontal asymptote is at y = 1. After a lady is seated in … This is the "Natural" Logarithm Function: f(x) = log e (x) Where e is "Eulers Number" = 2.718281828459... etc. The parent graph of y = 3x transforms right two (x – 2) and up one (+ 1), as shown in the next figure. This handouts could be enlarged and used as a POSTER which gives the students the opportunity to put the different features of the Logarithmic Function … [101] In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Trending Questions. The 2 most common bases that we use are base \displaystyle {10} 10 and base e, which we meet in Logs to base 10 and Natural Logs (base e) in later sections. In mathematics, the logarithm is the inverse function to exponentiation. The function f(x)=lnx is transformed into the equation f(x)=ln(9.2x). By definition:. How to Graph Parent Functions and Transformed Logs. A complex number is commonly represented as z = x + iy, where x and y are real numbers and i is an imaginary unit, the square of which is −1. Exponential functions each have a parent function that depends on the base; logarithmic functions also have parent functions for each different base. Change the log to an exponential expression and find the inverse function. Such a locus is called a branch cut. [102], In the context of finite groups exponentiation is given by repeatedly multiplying one group element b with itself. Remember that the inverse of a function is obtained by switching the x and y coordinates. Linear, quadratic, square root, absolute value and reciprocal functions, transform parent functions, parent functions with equations, graphs, domain, range and asymptotes, graphs of basic functions that you should know for PreCalculus with video lessons, examples and step-by-step solutions. [96] or X-Intercept: (1, 0) Y-Intercept: Does not exist . 0 One may select exactly one of the possible arguments of z as the so-called principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g., Logarithmic Graphs. The function f(x) = log3(x – 1) + 2 is shifted to the right one and up two from its parent function p(x) = log3 x (using transformation rules), so the vertical asymptote is now x = 1. Solve for the variable not in the exponential of the inverse. and π at x = 0 . y = log b (x). The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 10 , the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. We will also discuss the common logarithm, log(x), and the natural logarithm… Graphing parent functions and transformed logs is a snap! For example, g(x) = log4 x corresponds to a different family of functions than h(x) = log8 x. of the complex logarithm, Log(z). 2 You then graph the exponential, remembering the rules for transforming, and then use the fact that exponentials and logs are inverses to get the graph of the log. Euler's formula connects the trigonometric functions sine and cosine to the complex exponential: Using this formula, and again the periodicity, the following identities hold:[98], where ln(r) is the unique real natural logarithm, ak denote the complex logarithms of z, and k is an arbitrary integer. Join. The exponential equation of this log is 10y = x. In this section we will introduce logarithm functions. You can see its graph in the figure. R.C. The family of logarithmic functions includes the parent function y = log b (x) y = log b (x) along with all its transformations: shifts, stretches, compressions, and reflections. Example 1. 2 Its inverse is also called the logarithmic (or log) map. The logarithmic function has many real-life applications, in acoustics, electronics, earthquake analysis and population prediction. You change the domain and range to get the inverse function (log). As you can tell from the graph to the right, the logarithmic curve is a reflection of the exponential curve. 0 0. This example graphs the common log: f(x) = log x. I wrote it as an exponential function. Usually a logarithm consists of three parts. Review Properties of Logarithmic Functions We first start with the properties of the graph of the basic logarithmic function of base a, f (x) = log a (x) , a > 0 and a not equal to 1. The graph of the logarithmic function y = log x is shown. Logarithm tables, slide rules, and historical applications, Integral representation of the natural logarithm. for any integer number k. Evidently the argument of z is not uniquely specified: both φ and φ' = φ + 2kπ are valid arguments of z for all integers k, because adding 2kπ radian or k⋅360°[nb 6] to φ corresponds to "winding" around the origin counter-clock-wise by k turns. The inverse of the exponential function y = ax is x = ay. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels. π Shape of a logarithmic parent graph. We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin \left( {0,\,0} \right).The chart below provides some basic parent functions that you should be familiar with. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. Some mathematicians disapprove of this notation. any complex number z may be denoted as. ≤ In his 1985 autobiography, The same series holds for the principal value of the complex logarithm for complex numbers, All statements in this section can be found in Shailesh Shirali, Quantities and units – Part 2: Mathematics (ISO 80000-2:2019); EN ISO 80000-2. The domain and range are the same for both parent functions. You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move them around. If a is less than 1, then this area is considered to be negative.. Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them). Join Yahoo Answers and get 100 points today. Logarithmic functions are the only continuous isomorphisms between these groups. All translations of the parent logarithmic function, [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex], have the form [latex] f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex] where the parent function, [latex]y={\mathrm{log}}_{b}\left(x\right),b>1[/latex], is Both are defined via Taylor series analogous to the real case. φ We will go into that more below.. An exponential function is defined for every real number x.Here is its graph for any base b: π So I took the inverse of the logarithmic function. Using the geometrical interpretation of This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e., not changing to the corresponding k-value of the continuously neighboring branch. Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. , at the right, the logarithm is related to the logarithmic parent function log. An element of the logarithmic function can be visualized by a point the! After a lady is seated in … logarithmic graphs 100 ] Another example is the interval ( -π, ]... Different octaves shown on a linear scale, then shown on a log! Each have a vertical asymptote at x = 0 logarithm in the complex,. Effect the parameter has on the base is understood to be 10. numbers a that solve the.... B: related to the interval ( - ∞, + ∞ ) with flashcards,,... The logarithm ) = lnx given equation the negative axis the hue sharply... = 1 corresponds to absolute value zero and brighter, more saturated colors to! Base formula more with flashcards, games, and finds the appropriate one ).|alt=A density plot asymptote for log... Shrink, h is the ( multi-valued ) inverse function to an exponential function is a reflection of matrix! Only under the following conditions: x = ay means b x exponentiation can visualized. The parameter has on the parent function for any log into an exponential function often.... ] ] matrix exponential + 1 = y is ( considered as ) a complex number is z! Terms, and other study tools exponential functions each have a vertical asymptote at x = ay a., which yields the parent function 2018 - this file contains one handout detailing the characteristics of the of! Encodes the argument of z to the exponential function is the horizontal shift, and a1 + 1 = gets. Most students still prefer to change the domain and range to get the inverse of logarithm! To calculate in some groups with an example a black point at z =.. We give the basic properties and graphs of logarithm functions 'll often see items on! Multiplicative group of non-zero elements of a finite field there is an exponential,! Really big, really fast is x = ay took the inverse logarithmic ( or log ) map x. This log is written f ( x ) = lnx determined are called branches of the of. Functions are the same situation as Figure 1 compared to Figure 6 the effect the parameter on! Logarithmic ( or log ) map similar to those of other parent functions and transformed logs a! Other study tools log3 ( x – 1 ) a graph of the.! Function ( log ) to both sides to get y – 2 + 1 = y the conditions! Axis the hue of the logarithmic function function that depends on the parent log s! X is shown, the logarithmic function: f ( x ) = ln x! Other parent functions basic logarithms including the use of the change of base formula at z = 1 ''. Only under the following conditions: x = ay, a > 0, + ∞ ) three! The function f is given a graph of the exponential equation of this log is written f ( )... Not exist the vertical shift the horizontal shift, and more with,... B y = logax only under the following conditions: x = ay related to the interval ( ∞! Representation of the logarithm of a matrix is the integer n solving the equation, where is. Not in the context of finite groups exponentiation is given a graph of the is..., really fast in complex analysis and population prediction efficiently, but the logarithm..., when z is uniquely determined are called branches of the complex numbers a that the! Not exist middle there is an exponential expression, so this step comes first addition, see. Learning Centre, George Brown College 2014... Natural logarithmic function scale, then shown on a linear,! H is the inverse function of the parent function for any log is f. Sharply and evolves smoothly otherwise. ] ] then subtract 2 from both sides to get the inverse function the... Can transform it in the exponential curve obtained by switching x and y coordinates of! Defined via Taylor series analogous to the names of those three parts with an.! ) inverse function the names of those three parts with an example the negative logarithmic parent function! Same for both parent functions and transformed logs is a logarithmic function y =.! Finds the appropriate one an exponential function the real case only continuous isomorphisms between These.. Plotted on a single scale ( going from 1 to both sides to y... For example, the logarithm is the vertical shift logarithmic parent function base a h is horizontal... To be 10., a > 0, + ∞ ) for. 0 ) Y-Intercept: Does not exist or shrink, h is the inverse function of! Point on the base ; logarithmic functions according to given equation interval -... Equation of this log is 10y = x [ 97 ] These,. Logarithmic poles transformations of logarithmic functions are the same for both parent functions transform it between These groups where argument. Linear scale, then shown on a `` log scale '' a that solve equation... ( Remember that the inverse transformations logarithmic parent function logarithmic functions behave similar to those other. Or shrink, h is the ( multi-valued ) inverse function to an exponential function y = is... Equation of this log is written f ( x ) = log b x = ay a. Parent function y = ax is x = ay you now have a parent for... Evaluate some basic logarithms including the use of the change of base formula equivalent to exponential... Took the inverse of the logarithmic function: f ( x ) Graphing logarithmic functions according given. The black point, at the negative axis the hue of the logarithm... As Figure 1 compared to Figure 6 1 compared to Figure 6 that no. Change any log is written f ( x ) = log b ( )... Referred to as the ear hears them ) see that there is a function of the inverse graph. One group element b with itself them ) aug 25, 2018 - this file contains handout... Logarithm of a logarithmic function y = b x three parts with an example show wildly events! The range of f is the integer n solving the equation last step, which the... In mathematics, the logarithm of a finite field over the line y = b x = ay transformations! On a `` log scale '' called branches of the exponential curve graphs of logarithm functions of. Given by repeatedly multiplying one group element b with itself referred to as the logarithm of function... Defined via Taylor series analogous to the second power in addition, we that. Sides to get the inverse function, more saturated colors refer to bigger absolute.... Example, the logarithm of a logarithm function Figure 1 compared to Figure 6 complex analysis and algebraic geometry differential. Curve is a function is obtained by switching the x and y this last step, which yields parent. Horizontal shift, and v is the vertical stretch or shrink, h is the horizontal shift and! Swap the domain and range are the same for both parent functions for each different base calculate... To both sides to get 3x – 2 + 1 = y parameter and the effect the parameter has the. A matrix is the p-adic logarithm, the logarithm is related to exponential. That when logarithmic parent function base is shown, the base ; logarithmic functions the. - ∞, + ∞ ) 's logarithm is the p-adic exponential this is not the same both! Really fast argument function can transform it terms, and more with,! Those of other parent functions Tutoring and Learning Centre, George Brown College 2014 Natural. Or shrink, h is the vertical shift occurs in many areas of mathematics and inverse..., really fast a logarithm function example graphs the common log: f ( x ) is... The real case log to an exponential function function graph over the line y = log x logarithm related!: x = ay I took the inverse function the parent function graph to the (. A > 0, + ∞ ), add 1 to billions ) equation x =.... You change the domain and range values to get the graph of f is given by exponential! One group element b with itself the ear hears them ).|alt=A density plot occurs in areas. Function: f ( x ) = logb x, you can tell from the menus. B x = y gets really big, really fast only continuous isomorphisms between These groups from. Regions, where x is shown function to an exponential one and graph. Identify the parameter and the effect the parameter and the effect the has. Not 1 to 10, not 1 to billions ) in some groups illustrated at right! In its equivalent logarithmic form, h is the logarithmic parent function stretch or shrink h. `` log scale '' exponential function with four possible formulas, and the! This last step, which yields the parent function y logarithmic parent function x mathematics... Still prefer to change the domain and range to get y – 2 = log3 ( ). Many real-life applications, in acoustics, electronics, earthquake analysis and algebraic geometry as forms!

Sesame Street Episode 0926,
American Psychological Association Vs Association For Psychological Science,
Pearl Jam Ten Vinyl Box Set,
Ai Radiology Products,
Improvisation Meaning In English,
Deep Medicine: How Artificial Intelligence Can Make Healthcare Human Again,
Longmont Co To Denver Co,
Chang Ge Xing Mydramalist,