Warm-up: Solve for x. 2x = 8 2) 4x = 1 3) ex = e 4) 10x = 0.1 HW: Page 317 (2 – 18 even, 21, 22, 25, 26, 29, 30, 32-40 even, 51 – 61 odd) Warm-up: Solve for x. 2x = 8 2) 4x = 1 3) ex = e 4) 10x = 0.1 Describe how the graph of g is related to the graph of f. 5) g(x) = f(x + 2) 6) g(x) = -1 + f(x) 7) g(x) = -f(x) 8) g(x) = f(-x)

3.2 Logarithmic Functions and Their Graphs Objective: Graph logarithmic and natural log functions Apply properties of logarithms and natural logarithms

y = loga x if and only if x = a y. For x  0 and 0  a  1, y = loga x if and only if x = a y. The function given by f (x) = loga x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y A logarithm is an exponent! A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay

Equivalent Exponential Equation Examples: Evaluate each Logarithm. y = logax is equivalent to x = ay Solution Equivalent Exponential Equation Logarithm log216 16 = 2y 16 = 24  y = 4 log2( ) = 2 y = 2-1 y = –1 log1255 5 = 125y 5 = 1251/3  y = 1/3 log51 1 = 5y 1 = 50  y = 0

no power of 10 gives a negative number The base 10 logarithm function f (x) = log10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. Examples: Calculate the values using a calculator. Function Value Keystrokes Display log10 100 LOG 100 ENTER 2 log10( ) LOG ( 2 5 ) ENTER – 0.3979400 log10 5 LOG 5 ENTER 0.6989700 log10 –4 LOG –4 ENTER ERROR no power of 10 gives a negative number

3. loga ax = x and alogax = x inverse property Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. loga ax = x and alogax = x inverse property 4. If loga x = loga y, then x = y. one-to-one property Examples: Solve for x: log6 6 = x log6 6 = 1 property 2 x = 1 Simplify: log3 35 log3 35 = 5 property 3 Simplify: 7log79 7log79 = 9 property 3

horizontal asymptote y = 0 Graph f (x) = log2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function y = 2x in the line y = x. x y y = 2x y = x 8 3 4 2 1 –1 –2 2x x horizontal asymptote y = 0 y = log2 x x-intercept (1, 0) vertical asymptote x = 0

Example: Graph the common logarithm function f(x) = log10 x. 0.602 0.301 –1 –2 f(x) = log10 x 10 4 2 x y x 5 –5 by calculator f(x) = log10 x (1, 0) x-intercept x = 0 vertical asymptote

The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a  1) y-axis vertical asymptote x y Graph of f (x) = loga x (a  1) y = a x y = x range 1. domain y = log2 x 2. range 3. x-intercept (1, 0) 4. vertical asymptote domain x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x

Ex: Graph the logarithmic function f(x) = log10(x – 1) SHIFT f(x) = log10x to the RIGHT 1 y x 5 –5 f(x) = log10 (x – 1) (2, 0) x-intercept x = 1 vertical asymptote

Ex: Graph the logarithmic function f(x) = 2 + log10x SHIFT f(x) = log10x to the UP 2 f(x) = 2 + log10x y x 5 –5 (1, 2) x = 0 vertical asymptote

The function defined by f(x) = loge x = ln x 5 –5 y = ln x The function defined by f(x) = loge x = ln x (x  0, e 2.718281) is called the natural logarithm function. y = ln x is equivalent to e y = x Use a calculator to evaluate: ln 3, ln –2, ln 100 Function Value Keystrokes Display ln 3 LN 3 ENTER 1.0986122 ln –2 LN –2 ENTER ERROR ln 100 LN 100 ENTER 4.6051701

Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property inverse property property 2 property 1

Finding the Domains of Logarithmic Functions Every domain of every ln x is the set of positive real numbers. (ln (-1)) is undefined. a. f(x) = ln(x – 2) b. g(x) = ln(2 – x) c. h(x) = lnx2 x – 2 > 0 2 – x > 0 x2 > 0 D: (2, ) D: (-, 2) D: All Reals except x = 0

Sneedlegrit: Find the domain, vertical asymptote, and x-intercept of f(x) = log4(x – 3) then sketch its graph. HW: Page 317 (2 – 18 even, 21, 22, 25, 26, 29, 30, 32-40 even, 51 – 61 odd)