Intermediate Algebra (11th edition)

Some graphing calculators have the capability of drawing the inverse of a function. For

example, the two screens that follow show the graphs of and. The

graph of g was obtained by drawing the graph of ,since. (Compare to

FIGURE 9in this section.)

ƒ-^1 g 1 x 2 =ƒ-^11 x 2

ƒ 1 x 2 = 2 x g 1 x 2 =log 2 x

SECTION 10.4 Properties of Logarithms 595

10

–10

–10 10

f(x) = 2x

10

–10

–10 10

g(x) = log 2 x

Use a graphing calculator with the capability of drawing the inverse of a function to draw the

graph of each logarithmic function. Use the standard viewing window.

67.

(Compare to Exercise 49.)

68.

(Compare to Exercise 50.)

g 1 x 2 =log 3 x g 1 x 2 =log 5 x

69.

(Compare to Exercise 51.)

70.

(Compare to Exercise 52.)

g 1 x 2 =log1/3 x g 1 x 2 =log1/5 x

Simplify each expression. Write answers using only positive exponents. See Section 5.1.

71. 
    
    
    
    72. 
        
        
        
        73. 
            
            
            
            74. 1932 -^2
            
            
            
            
        
        
        
        
    
    
    
    

78

7 -^4

5 -^3

58

47 # 42

PREVIEW EXERCISES

OBJECTIVES

Properties of Logarithms

10.4

1 Use the product

rule for logarithms.

2 Use the quotient

rule for logarithms.

3 Use the power rule

for logarithms.

4 Use properties to

write alternative

forms of

logarithmic

expressions.

Logarithms were used as an aid to numerical calculation for several hundred years.

Today the widespread use of calculators has made the use of logarithms for calcula-

tion obsolete. However, logarithms are still very important in applications and in fur-

ther work in mathematics.

OBJECTIVE 1 Use the product rule for logarithms. One way in which loga-

rithms simplify problems is by changing a problem of multiplication into one of

addition. We know that and

This is an example of the following rule.

log 2 14 # 82 = log 2 4 + log 2 8 32 = 4 # 8

log 2 32 = log 2 4 + log 2 8 5 = 2 + 3

log 2 4 =2,log 2 8 =3, log 2 32 =5.

Product Rule for Logarithms

If x, y, and bare positive real numbers, where then the following is true.

That is, the logarithm of a product is the sum of the logarithms of the factors.

logb xylogb xlogb y

bZ1,