 # Objectives Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions.

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Objectives Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions.

How many times would you have to double \$1 before you had \$8?
You could use an exponential equation to model this situation. 1(2x) = 8. You may be able to solve this equation by using mental math if you know 23 = 8. So you would have to double the dollar 3 times to have \$8.

How many times would you have to double \$1 before you had \$512?
You could solve this problem if you could solve 2x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value.

You can write an exponential equation as a logarithmic equation and vice versa.
Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent. Reading Math

Example 1: Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 35 = 243 25 = 5 104 = 10,000 6–1 = ab = c The base of the exponent becomes the base of the logarithm. log3243 = 5 1 2 1 2 log255 = The exponent is the logarithm. log1010,000 = 4 1 6 log = –1 1 6 An exponent (or log) can be negative. logac =b The log (and the exponent) can be a variable.

Write each exponential equation in logarithmic form. 92= 81 33 = 27
Check It Out! Example 1 Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 92= 81 33 = 27 x0 = 1(x ≠ 0) The base of the exponent becomes the base of the logarithm. a. log981 = 2 b. log327 = 3 The exponent of the logarithm. The log (and the exponent) can be a variable. c. logx1 = 0

Example 2: Converting from Logarithmic to Exponential Form
Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation log99 = 1 log = 9 log82 = log = –2 logb1 = 0 The base of the logarithm becomes the base of the power. 91 = 9 The logarithm is the exponent. 29 = 512 1 3 1 3 8 = 2 A logarithm can be a negative number. 1 1 16 4–2 = 16 Any nonzero base to the zero power is 1. b0 = 1

Write each logarithmic form in exponential equation.
Check It Out! Example 2 Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation log1010 = 1 log12144 = 2 log 8 = –3 The base of the logarithm becomes the base of the power. 101 = 10 122 = 144 The logarithm is the exponent. 1 2 –3 = 8 An logarithm can be negative. 1 2

A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example.

A logarithm with base 10 is called a…. common logarithm.
If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log105. You can use mental math to evaluate some logarithms.

Example 3A: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math. log 0.01 -2 = ________ 10x = 0.01 The log is the exponent. 10–2 = 0.01 Think: What power of 10 is 0.01? log 0.01 = –2

Example 3B: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math. log5 125 3 = ________ 5x = 125 The log is the exponent. 53 = 125 Think: What power of 5 is 125? log5125 = 3

Example 3C: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math. 1 5 log5 -1 = ________ 5x = 1 5 The log is the exponent. 5–1 = 1 5 1 5 Think: What power of 5 is ? log = –1 1 5

-5 Check It Out! Example 3a Evaluate by using mental math. log 0.00001
= ________ 10x = The log is the exponent. 10–5 = Think: What power of 10 is 0.01? log = –5

-1 Check It Out! Example 3b Evaluate by using mental math. log250.04
= ________ 25x = 0.04 The log is the exponent. 25–1 = 0.04 Think: What power of 25 is 0.04? log = –1

Exponential and logarithmic operations undo each other since they are inverse operations.

Example 4: Recognizing Inverses
Simplify each expression. a. log3311 b. log381 c. 5log510 log3311 log334 5log510 10 4 11

log 100.9 2log2(8x) 0.9 8x Check It Out! Example 4
a. Simplify log100.9 b. Simplify 2log2(8x) log 100.9 2log2(8x) 0.9 8x

Because logarithms are the inverses of exponents, the inverse of an exponential function, such as
y = 2x, is a logarithmic function, such as y = log2x. You may notice that the domain and range of each function are switched. Domain Range

1. Change 64 = 1296 to logarithmic form. log61296 = 4
Lesson Quiz: Part I 1. Change 64 = 1296 to logarithmic form. log61296 = 4 27 = 9 2 3 2. Change log279 = to exponential form. 2 3 Calculate the following using mental math. 3. log 100,000 5 4. log648 0.5 5. log3 1 27 –3

Lesson Quiz: Part II 6. Use the x-values {–2, –1, 0, 1, 2, 3} to graph f(x) =( )X. Then graph its inverse. Describe the domain and range of the inverse function. 5 4 - 2 -1 1 2 3 - 2 .64 -1 .8 1 1.25 2 1.5625 3 1.9531 D: {x > 0}; R: all real numbers

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