CHAPTER 44-3 logarithmic functions

objectivesWrite equivalent forms for exponential and logarithmic functions.

Write, evaluate, and graph logarithmic functions.

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.

Logarithmic functions

You can write an exponential equation as a logarithmic equation and vice versa.

Example #1

◦Write each exponential equation in logarithmic form.Exponential

EquationLogarithmic

Form

35 = 243

25 = 5

104 = 10,000

6–1 =

ab = c

Example #2

◦Write each exponential equation in logarithmic form.

Exponential Equation

Logarithmic Form

92= 81

33 = 27

x0 = 1(x ≠ 0)

Student guided practice◦Do problems 2 to 5 in your book page 253

Example #3

◦Write each logarithmic form in exponential equation.Logarithmic

FormExponential

Equation

log99 = 1

log2512 = 9

log82 =

log4 = –2

logb1 = 0

Student guided practice◦Do problems 6 to 9 in your book page253

Logarithmic functions

◦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.

Example #4

◦Evaluate by using mental math.

log 0.01

Example #5

◦Evaluate by using mental math.

log5 125

Example #6

◦Evaluate by using mental math.

log5

15

Student guided practice◦Do problems 10 to13 in your book page 253

Properties of logarithms◦ 1. loga (uv) = loga u + loga v

◦ 2. loga (u / v) = loga u - loga v

◦ 3. loga un = n loga u

◦Many logarithmic expressions may be rewritten, either expanded or condensed, using the three properties above. Expanding is breaking down a complicated expression into simpler components. Condensing is the reverse of this process.

Properties of logs◦ In less formal terms, the log rules might be expressed as:

◦ 1) Multiplication inside the log can be turned into addition outside the log, and vice versa.

◦ 2) Division inside the log can be turned into subtraction outside the log, and vice versa.

◦ 3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.

Example ◦ Expand log3(2x).

Expand log4( 16/x ).

Student guided practice◦Do problems 1-6 in the worksheet

Condensing logs◦Simplify log2(x) + log2(y).

◦Simplify log3(4) – log3(5).

◦Simplify 3log2(x) – 4log2(x + 3) + log2(y).

Student guided practice◦Do odd problems from 13-20 in the worksheet

Graphs of log functions

◦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.

The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R).

Example #7

◦Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function.

1f(x) = 1.25x

210–1–2x

0.64 0.8 1.25 1.5625

Continue◦ Inverse

210–1–2f–1(x) = log1.25x

1.56251.2510.80.64x

Example #8

◦Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function.

f(x) = x 1 2

x –2 –1 0 1 2

f(x) =( ) x 4 2 1

Continue◦ Inverse

Student guided practice◦Do problems 14 and 15 in your book page 253

Homework◦Do problems 17-25 in your book page 253

Closure◦ Today we learned about logarithmic functions

◦Next class we are going to have. A quiz and review

Have a great day