6. 3 Logarithmic FunctionsObjectives: Write equivalent forms

for exponential and logarithmic equations.

Use the definitions of exponential and logarithmic functions to solve

equations.Standard: 2.8.11.S. Analyze

properties and relationships of functions.

Logarithms are used to find unknown exponents in exponential models.

Logarithmic functions define many measurement scales in the sciences, including the pH, decibel, and Richter scales.

With logarithms, you can write an exponential equation in an equivalent logarithmic form.

For any positive base b, where b ≠ 1 if and only if x = log y

Ex 1.

a. Write in logarithmic form. ________________________

b. Write in exponential form. _______________________

c. Write 112 = 121 in logarithmic form. _________________________

d. Write log 6 36 = 2 in exponential form. _______________________

e. Write 7-2=1/49 in logarithmic form. __________________________

f. Write log 3 1/81= -4 in exponential form. _______________________

2 = log11 121

62 = 36

Log7 (1/49) = -2

3-4 = 1/81

You can evaluate logarithms with a base of 10 by using the log key on a calculator.

Ex 2. Solve each equation for x. Round your answer to the nearest thousandth.

a). 10x= 1/109 b). x = log101/109 x = -2.037

c). 10x= 1.498 d). 10x= 7210 x = log10 1.498 x = log107210 x = .176 x = 3.858

The inverse of the exponential function y = 10x is x = 10y.

To rewrite x = 10y in terms of y, use the equivalent logarithmic form, y = log 10 x.

Examine the tables & graphs below to see the inverse relationship between y=10x and y = log10x.

y= 10x

y=x

y = log10x

BeBelow summarizes the relationship between the domain and range of y = 10x and of y = log10 X.

• y = 10x

Domain: all Real #s

Range: all positive Real #s

• y = log10 X

Domain: all positive real #s

Range: all Real #s

The logarithmic function y = log b x withbase b, or x = by, is the inverse of the exponential function y = bx, where b ≠ 1 and b > 0.

One-to-one Property of ExponentsIf bx = by, then x = y.

Ex. 3 Find the value of v in each equation.

B.A.

d. v = log464

4v = 64

4v = 43 (same base)

v = 3

e. 2 = logv25 v2 = 25 v2=52

v = 5

f. 6 = log3v v = 36

v = 729

g. v = log10 1000 10v = 1000 10v = 103

v = 3

h. 2 = log7V V = 72

V = 49

I. 1 = log3v 31 = v 3 = v

HomeworkPg. 374-375 #12-84 even