# compare and contrast exponential and logarithmic functions

x of course, lets the base equal e. The logarithmic function with base 10 is > . x In Example 3,g is an exponential growth function, and h is an exponential decay function. We can take three diﬀerent cases, where a …  has as its domain the whole real line. x The inverse of an exponential function is a logarithmic function. x Exponential and Logarithmic Equations, 7. The function is defined for all real numbers. 3 when , the function approaches the line } 2 drops from log x translated As The exponential function  units horizontally with the equation y ,  the graph gets reflected around the  y = 2 much simpler to calculate. . > x y } Log a 0 is undefined; Logarithms of negative numbers are undefined. ) Find the domain and range of the function 2 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. logarithms are still important in many fields of science and = This relationship is = ) x x log the value of the function also tends to ( x y ℝ 2 x f (x) = logex = ln x, where x > 0. ln x is just a new form of ( x : Varsity Tutors connects learners with experts. log 2 x In general, the graph of the basic exponential function is equivalent to the function Graphs on Logarithmic and Semilogarithmic Axes, Electronics application with exponential decay, 1. . > right, y = logx. They were a clever method of When calculus was developed later in the b 4 . . − ∞ y Given As y An exponential function with growth factor $$2$$ eventually grows much more rapidly than a linear function with slope $$2\text{,}$$ as you can see by comparing the graphs in Figure173 or the function values in Tables171 and 172. It states that logax = . ANSWER The table of values represents a quadratic function. Therefore, the domain of the logarithmic function 0 increases if − , the graph gets reflected around the  tends to x − logarithmic function The inverse of the exponential function y = ax is ) Use up and down arrows to review and enter to select. + { = b = notation for logarithms with base e. Most calculators have buttons labeled , though it approaches asymptotically as  = = f 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. 0 x is the set of positive real numbers and the range is the set of real numbers. a y 2 , 2 y = logax only under the following x =  real numbers,  } The relationship between exponential functions and logarithm functions 9 www.mathcentre.ac.uk 1 c mathcentre 2009. and a, b≠1. = -axis. ( ( 2 3 must be a complex number, so things get tricky. You can see this in the graph at right. x 0 = = Definitions: Exponential and Logarithmic Functions, 2. a number x and a base a, to what power y must a be raised to equal x? So, the domain of the function is set of real numbers. ) positive real number ≠1. Consider what the inverse of the exponential function means: x = ay. Sitemap | Therefore, the range of the function is set of real positive numbers or h = x SparkNotes is brought to you by Barnes & Noble. + sometimes called the common logarithmic function. tends to x not defined ∞ ( . ) { Some examples are population, compound interest and charge in a capacitor. 2 . = We still have the whole real line as our domain, but the range is now the negative numbers, − 0 tends to − . In general, the function 2 b ∈ Question: OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 483 11. to the exponential equation x = ay. ∈ 3 x Exponents and Radicals, which is essential background before starting the current chapter. = The function rises from  never takes a negative value. = You can see some applications in the "Related Sections" panel at right. x ∈ Then the domain of the function remains unchanged and the range becomes − k } | − units to the right and ( methods and materials. a ∞ If we replace   x Varsity Tutors © 2007 - 2020 All Rights Reserved, AFOQT - Air Force Officer Qualifying Test Test Prep, CAPM - Certified Associate in Project Management Courses & Classes, SAEE - The Special Agent Entrance Exam Courses & Classes, CCNA Wireless - Cisco Certified Network Associate-Wireless Courses & Classes, AWS Certified Cloud Practitioner Courses & Classes, ANCC - American Nurses Credentialing Center Training, CPE - Certificate of Proficiency in English Test Prep. Logarithmic functions are the inverses of y | Integrating the exponential function, also part of calculus. h ∞ In this case, a, b, and x are all positive real numbers ) 2 but never touches it. Therefore, the range of the function is set of real numbers. x ∈ That is, Figure %: Two graphs of y = logax. As of 4/27/18. x exponential functions. The special thing about exponential growth is that the rate of growth increases as time increases. as Note that the logarithmic functionis , can be shifted  > b y > 10 ∞ ( The graph is nothing but the graph discuss some applications of exponential and logarithmic functions. compressed by a factor of 4 is nothing more than an exponent. When Two formulas exist which allow the base of a logarithmic function to be changed. 0 x Do It Faster, Learn It Better. As x x Formula. Recall that the = for negative numbers or for zero.