Math 1431 Section 4.3 Section 4.3: Logarithmic Function The exponential function is 1-1; therefore, it has an inverse function. The inverse function of the exponential function with base a is called the logarithmic function with base a. The function fx( ) loga x is the logarithmic function with base a with x > 0, a > 0 and a 1. For example, for a > 1: If a = 10 then fx() log x. This is known as the common base. If a = e then fx() ln x. This is known as the natural logarithm. ln(0) is undefined. ln(1) 0 Graphs The orientation of the logarithmic function depends on its base, 0 < a < 1 or a > 1. Both graphs have a vertical asymptote of x = 0 (the y-axis). Their domain is 0, and range is , . 1 Math 1431 Section 4.3 Example 1: Find the domain of yx ln(ln( )) . The logarithmic function with base a satisfies: x loga ax for all x , and log x axa , for all x 0, Examples: 3 log2=2 ln e = eln 5 = 3log3 x = 2 Math 1431 Section 4.3 Converting Expressions y Fact: y loga x is equivalent to ax For example: 5 2 32 is equivalent to log2 32 5 2 log4 16 2 is equivalent to 16 4 Laws of Logarithms Product Rule for Logs: logaaa x yxy log log x Quotient Rule for Logs: logaaa logx log y y p Power Rule for Logs: logaa x px log Example 2: Expand using the Laws of Logarithms: x3y ln zx1 1 2 Example 3: Simplify using the Laws of Logarithms: logx logxz 1 4log 3 3 Math 1431 Section 4.3 Differentiating Natural Logarithmic Function: d 1 ln x dx x If u is a function of x , then use the chain rule. Example 4: Differentiate the following functions. a. yxxln 43 5 b. yx ln sin c. f xx ln ln 2 Sometimes it’s easier to use the rules of logarithms to differentiate a log function. 4 Math 1431 Section 4.3 x4 Example 5: Differentiate . Then find the equation of the tangent line at the point gx() ln2 31x 1 1, ln . 2 5 Math 1431 Section 4.3 Example 6: Determine where f ()xxx ln is increasing/decreasing, any relative extrema, concavity, and any points of inflection. 6 Math 1431 Section 4.3 Different Bases ln x Change of base formula: log x a ln a d 11 log x dxa xln a If u is a function of x , then use the chain rule. Example 7: Find the derivative of each function. a. yx log2 3 b. yxxlog3 4 2 7 Math 1431 Section 4.3 log x c. fx() 9 x2 We know how to find the derivative of functions such as: x5 ---use the power rule 5x ---use exponential rule But how about xx ? We will use a method called Logarithmic Differentiation. 8 Math 1431 Section 4.3 Example 8: Find the derivative of yx tan x . Step 1: Take natural logs of both sides of the equation. Step 2: Use any rules of logs to simplify the equation. Step 3: Take the derivative of both sides of the equation. Step 4: Solve for y ' . Remember what y was equal to originally. 9 Math 1431 Section 4.3 Example 9: Find the derivative of yx 21x . Step 1: Take natural logs of both sides of the equation. Step 2: Use any rules of logs to simplify the equation. Step 3: Take the derivative of both sides of the equation. Step 4: Solve for y ' . Remember what y was equal to originally. 10 Math 1431 Section 4.3 Try these: Find the derivative of: yx log cos 4 . Let f xx ln cos 2 x , find f ' . 2 x Determine where fx() 2 x ln is increasing/decreasing. 4 2 x Find the points of inflection for the function fx() 4 x ln. 4 2x Find the derivative of f xe ln(2 x ) . x2 x Find the derivative of fx ln 5 . 4sin x Find the slope of the tangent line to the curve yx2cos at x 2 . 11 .