5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and graph natural logs Use logarithmic functions to model and solve real-life problems.

Is this function one to one? Must pass the horizontal line test. f(x) = 3x Is this function one to one? Yes Does it have an inverse? Yes

Logarithmic Function of base “a” Definition: Logarithmic function of base “a” - For x > 0, a > 0, and a  1, y = logax if and only if x = ay Read as “log base a of x” f(x) = logax is called the logarithmic function of base a.

The most important thing to remember about logarithms is…

a logarithm is an exponent.

Therefore, all logarithms can be written as exponential equations and all exponential equations can be written as logarithmic equations.

Write the logarithmic equation in exponential form 34 = 81 163/4 = 8 Write the exponential equation in logarithmic form 82 = 64 4-3 = 1/64 log 8 64 = 2 log4 (1/64) = -3

Evaluating Logs 2y = 32 2y = 25 y = 5 Think: y = log232 f(x) = log232 Step 1- rewrite it as an exponential equation. f(x) = log42 4y = 2 22y = 21 y = 1/2 2y = 32 f(x) = log10(1/100) Step 2- make the bases the same. 10y = 1/100 10y = 10-2 y = -2 2y = 25 f(x) = log31 Therefore, y = 5 3y = 1 y = 0

Evaluating Logs on a Calculator You can only use a calculator when the base is 10 Find the log key on your calculator.

Why? log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!! Evaluate the following using that log key. log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!! Why?

Properties of Logarithms loga1 = 0 because a0 = 1 logaa = 1 because a1 = a logaax = x and alogax = x If logax = logay, then x = y

Simplify using the properties of logs Rewrite as an exponent 4y = 1 Therefore, y = 0 log41= log77 = 1 Rewrite as an exponent 7y = 7 Therefore, y = 1 6log620 = 20

Use the properties of logs to solve these equations. log3x = log312 x = 12 log3(2x + 1) = log3x 2x + 1 = x x = -1 log4(x2 - 6) = log4 10 x2 - 6 = 10 x2 = 16 x = 4

Review: How do you find the inverse of a function? Application of what you know… What is the inverse of f(x) = 3x? y = 3x x = 3y y = log3x f-1(x) = log3x Rewrite the exponential as a logarithm…

Find the inverse of the following exponential functions… f(x) = 2x f-1(x) = log2x f(x) = 2x+1 f-1(x) = log2x - 1 f(x) = 3x- 1 f-1(x) = log3(x + 1)

Find the inverse of the following logarithmic functions… f(x) = log4x f-1(x) = 4x f(x) = log2(x - 3) f-1(x) = 2x + 3 f(x) = log3x – 6 f-1(x) = 3x+6

Graphs of Logarithmic Functions Graph g(x) = log3x It is the inverse of y = 3x Therefore, the table of values for g(x) will be the reverse of the table of values for y = 3x. y = 3x x y -1 1/3 1 3 2 9 y= log3x x y 1/3 -1 1 3 9 2  Domain? (0,)   Range? (-,)  Asymptotes? x = 0

Graphs of Logarithmic Functions g(x) = log4(x – 3) What is the inverse exponential function? y= 4x + 3 Show your tables of values.    y= 4x + 3 x y -1 3.25 4 1 7 2 19 y= log4(x – 3) x y 3.25 -1 4 7 1 19 2 Domain? (3,) Range? (-,) Asymptotes? x = 3

Graphs of Logarithmic Functions g(x) = log5(x – 1) + 4 What is the inverse exponential function? y= 5x-4 + 1  Show your tables of values.   y= 5x-4 + 1 x y 3 1.2 4 2 5 6 26 y= log5(x – 1) + 4 x y 1.2 3 2 4 6 5 26 Domain? (1,) Range? (-,) Asymptotes? x = 1

Natural Logarithmic Functions The function defined by f(x) = logex = ln x, x > 0 is called the natural logarithmic function.

Evaluating Natural Logs on a Calculator Find the ln key on your calculator.

Why? ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = 2.3321 ln -1 = ERROR!!! Evaluate the following using that ln key. ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = 2.3321 ln -1 = ERROR!!! Why?

Properties of Natural Logarithms ln1 = 0 because e0 = 1 Ln e = 1 because e1 = e ln ex = x and eln x = x If ln x = ln y, then x = y

Use properties of Natural Logs to simplify each expression Rewrite as an exponent ey = 1/e ey = e-1 Therefore, y = -1 ln 1/e= -1 2 ln e = 2 Rewrite as an exponent ln e = y/2 e y/2 = e1 Therefore, y/2 = 1 and y = 2. 5 eln 5=

Graphs of Natural Log Functions g(x) = ln(x + 2) Show your table of values. y= ln(x + 2) x y -2 error -1 .693 1 1.099 2 1.386   Domain? (-2,)   Range? (-,) Asymptotes? x = -2

Graphs of Natural Log Functions g(x) = ln(2 - x) Show your table of values. y= ln(2 - x) x y 2 error 1 .693 -1 1.099 -2 1.386   Domain? (-2,)   Range? (-,) Asymptotes? x = -2