Chapter 1 | Functions and Graphs
105
The solution is x =10 4/3 =10 10 3 . c. Using the power property of logarithmic functions, we can rewrite the equation as ln(2 x )−ln ⎛ ⎝ x 6 ⎞ ⎠ =0. Using the quotient property, this becomes ln ⎛ ⎝ 2 x 5 ⎞ ⎠ =0.
Therefore, 2/ x 5 =1, which implies x = 2 5 .
We should then check for any extraneous solutions.
Solve ln ⎛
⎞ ⎠ −4ln( x ) =1.
⎝ x 3
1.31
When evaluating a logarithmic function with a calculator, you may have noticed that the only options are log 10 or log, called the common logarithm , or ln , which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base b . If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.
Rule: Change-of-Base Formulas Let a >0, b >0, and a ≠1, b ≠1. 1. a x = b x log b a
for any real number x . If b = e , this equation reduces to a x = e
x log e a = e x ln a .
log b x log b a
2. log a x = for any real number x >0. If b = e , this equation reduces to log a x = ln x ln a .
Proof For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base b >0, b ≠1, log b ( a x ) = x log b a . Therefore,
log b ( a x )
x log b a .
b
= b
In addition, we know that b x and log
b ( x ) are inverse functions. Therefore,
log b ( a x )
= a x .
b
x log b a .
Combining these last two equalities, we conclude that a x = b
To prove the second property, we show that
(log b a ) · (log a x ) = log b x . Let u = log b a , v = log a x , and w = log b x . We will show that u · v = w . By the definition of logarithmic functions, we
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