Math 112 Steiner sec 4 3 page 1 of 6 4 3 Logarithmic Functions Exponential functions and logarithmic functions are inverses of each other Remember exponential functions are in the form bx y If we swap the x s and y s we get the function b y x This can be re written in a new form called logarithmic log b x y Definition of the Logarithmic Function remember your book likes the variable a for base Let a be a positive number with a 1 The logarithmic function with base a denoted by log a x is defined by Note The answer to a logarithm is a power log a x y ay x Calculator functions log x is the same as log10 x with base 10 ln x is the same as ln e x with base e Logarithms and exponential conversions logarithms are exponential functions Exponent Form Logarithmic Form b y x e y x ln x y log b x y 10 y x log x y base power something log base something power Example 1 Write each equation in logarithmic form a 52 25 b x y z c e7 3 d ez k Example 2 Write each equation in exponential form a log2 8 3 b b log a 2 c ln 5 2 d log x d Math 112 Steiner sec 4 3 page 2 of 6 Remember the answer to a log problem is a power First we are going to evaluate logs without a calculator Example 3 Evaluate each without a calculator Do not give decimal answers a log2 x 3 b log x 2 c ln x 0 d log3 x 2 e log x f log x 4 1 2 9 1 2 Properties of General Logarithmic Common Logarithmic and Natural Logarithmic General Logarithmic Common Logarithmic base 10 Natural Logarithmic base e logb 1 0 1 log 1 0 1 1 ln 1 0 2 ln e 1 3 ln e x x 4 b e ln x x logb b 1 2 log 10 1 2 logb b x x 3 log 10 x x 3 logb x x 4 10log x x 4 Math 112 Steiner sec 4 3 page 3 of 6 Example 4 Evaluate each expression without using a calculator a log5 25 b ln 1 c ln e5 RULE If we have 2 exponential equations equal with the same bases then their powers must be equal If b x b y then x y Example 5 Evaluate each expression without using a calculator 1 e a log5 b ln c log 8 Example 6 Use the calculator to evaluate each of the following Round to 4 decimal places a log 50 b log 6 2 c ln 3 d ln 2 3 Math 112 Steiner sec 4 3 page 4 of 6 e ln Graphs of Logarithmic Functions Logarithmic functions and exponential functions are INVERSES Since the 2 functions are inverses of each other if we graphed them on the same coordinate system they would have symmetry of y x These 2 functions are graphed below Graph of exponential and logarithmic functions f x ex g x ln x h x x 4 2 2 5 Recall that for y a x Domain Range 0 Intercept 0 1 By looking at the graph you should notice that the domain of the log functions have been restricted You cannot take the log of 0 or a negative number Horizontal Asymptote y 0 Characteristics of f x loga x Math 112 Steiner sec 4 3 page 5 of 6 Domain 0 Range x intercept 1 0 Vertical Asymptote x 0 We can still use transformations to graph logs just like we have for other equations Example 7 Use transformations to graph each State the domain range and asymptote a g x 2 log5 x Domain Range Asymptote b g x log Domain Range Asymptote IMPORTANT We have to be very aware of the domain of logarithms We cannot take the log of 0 or a negative We will have restricted domains Example 8 Find the domain of the following functions a f x log2 x b f x log2 x c f x log x 5 domain domain Domain Math 112 Steiner sec 4 3 page 6 of 6 d f x ln x2 4 domain