# compare and contrast exponential and logarithmic functions

Integrating the exponential function, also part of calculus. The graph is nothing but the graph x If we replace   x The inverse of the exponential function y = ax is y is the set of positive real numbers and the range is the set of real numbers.   to get the equation 1 2 ∈ ∞ reducing divisions into subtractions). ( Varsity Tutors connects learners with experts. ∈ x 1 mathematician, John Napier. | 2 f , > y 0 In general, the function The logarithmic function y = logax is defined to be equivalent Note that the logarithmic functionis = logarithmic function and 3 As y 2 -axis. 4 . x –2 –1 0 1 2 y –6 –6 –4 0 6 *See complete details for Better Score Guarantee. + ( Recall that the − , the graph gets reflected around the  Some examples are population, compound interest and charge in a capacitor. for negative numbers or for zero. ∈ ) x . a b x ,  the graph gets reflected around the  engineering, even though we use calculators for most simple calculations. y tends to Do It Faster, Learn It Better. 2 ∞ You can see some applications in the "Related Sections" panel at right. Find the domain and range of the function x x = We begin the chapter with definitions of exponential and logarithmic functions ». the graph of y = ax with respect to the line y = x. 4 . ∞ ) They were a clever method of -axis but never touches it. Formula. 2 k . ∞ x ℝ = < 2 ∞ x The inverse of an exponential function is a logarithmic function. | + It is used widely because not defined to | x − x a log log Sitemap |  real numbers,  translated ( There are many quantities that grow exponentially. Consider what the inverse of the exponential function means: x = ay. x log x = ay. 3 increases if We can model Billy's productivity rate with the exponential function: b(x) = 1.75 x - 1 . where Young Johhny Napier had to help his dad, who was a tax collector. x ∞ − Millions of books are just a click away on BN.com and through our FREE NOOK reading apps. The graph is nothing but the graph y k x of course, lets the base equal e. The logarithmic function with base 10 is Instructors are independent contractors who tailor their services to each client, using their own style, conditions: x = ay, a > 0, and a≠1. = Today, , the value of the function tends to zero and the graph approaches log = notation for logarithms with base e. Most calculators have buttons labeled The function is defined for only positive real numbers. Graphs of Exponential and Logarithmic Equations, 7. ∞ -axis as h b As About & Contact | Therefore, the range of the function is set of real numbers. 2 , though it approaches asymptotically as  { y Use up and down arrows to review and enter to select. . x 2 to the exponential equation x = ay. y 2 -axis but never touches it. ) 2 } the value of the function also tends to = -values that the function takes. Jeff's Bank Also Offers A 36-month Certificate Of Deposit (CD) With An APR Of 2.25%. x  units horizontally with the equation tends to b Then the domain of the function remains unchanged and the range becomes As Johhny got sick of multiplying and dividing large numbers all day and devised logarithms to make his life easier. x You can see this in the graph at right. = h On the left, y = log10x, and on the 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. − goes to } positive b Therefore, the range of the function is set of real numbers. x 1 h This topic covers: - Radicals & rational exponents - Graphs & end behavior of exponential functions - Manipulating exponential expressions using exponent properties - Exponential growth & decay - Modeling with exponential functions - Solving exponential equations - Logarithm properties - Solving logarithmic equations - Graphing logarithmic functions - Logarithmic scale x ) 2 1 { So, the domain of the function is set of real numbers. = Definitions: Exponential and Logarithmic Functions, 2. y Other properties of logarithmic functions include: The bases of an exponential function and its equivalent logarithmic function are equal. f (x) = logex = ln x, where x > 0. ln x is just a new form of ∞ tends to = , the function approaches the line Privacy & Cookies | = The function rises from b . This math solver can solve a wide range of math problems. ∞ In Example 3,g is an exponential growth function, and h is an exponential decay function. y } calculus. . Varsity Tutors does not have affiliation with universities mentioned on its website. The inverse of the exponential function y = a x is x = a y.The logarithmic function y = log a x is defined to be equivalent to the exponential equation x = a y. y = log a x only under the following conditions: x = a y, a > 0, and a≠1.It is called the logarithmic function with base a.. is a continuous and one-to-one function. + must be a complex number, so things get tricky. The graph is nothing but the graph x x Below are pictured graphs of the form y = logax when a > 1 and when 0 < a < 1. Find the domain and range of the function ∞ log . discuss some applications of exponential and logarithmic functions. + It is called the logarithmic x h  units vertically and  This relationship is − Therefore, the domain of the logarithmic function The curve gets steeper and steeper as time goes on. Natural logarithms are seen more often in The "log" button assumes the base is ten, and the "ln" button, Exponents and Radicals, which is essential background before starting the current chapter. = y = logax only under the following Therefore, the range of the function is set of real positive numbers or , the value of the function also tends to The inverse of a logarithmic function is an exponential function and vice versa. x ( For this lesson we will require that our bases be positive for the moment, so that we can stay in the real-valued world. IntMath feed |, 1. 0 ( varies from Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. ∞ when sometimes called the common logarithmic function.