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Write the following exponential equations in logarithmic form.

  1. 3 2 = 9
  2. 5 3 = 125
  3. 2 1 = 1 2
  1. 3 2 = 9 is equivalent to log 3 ( 9 ) = 2
  2. 5 3 = 125 is equivalent to log 5 ( 125 ) = 3
  3. 2 1 = 1 2 is equivalent to log 2 ( 1 2 ) = 1
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Evaluating logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log 2 8. We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know 2 3 = 8 , it follows that log 2 8 = 3.

Now consider solving log 7 49 and log 3 27 mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know 7 2 = 49. Therefore, log 7 49 = 2
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know 3 3 = 27. Therefore, log 3 27 = 3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log 2 3 4 9 mentally.

  • We ask, “To what exponent must 2 3 be raised in order to get 4 9 ? ” We know 2 2 = 4 and 3 2 = 9 , so ( 2 3 ) 2 = 4 9 . Therefore, log 2 3 ( 4 9 ) = 2.

Given a logarithm of the form y = log b ( x ) , evaluate it mentally.

  1. Rewrite the argument x as a power of b : b y = x .
  2. Use previous knowledge of powers of b identify y by asking, “To what exponent should b be raised in order to get x ?

Solving logarithms mentally

Solve y = log 4 ( 64 ) without using a calculator.

First we rewrite the logarithm in exponential form: 4 y = 64. Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know

4 3 = 64

Therefore,

log ( 64 ) 4 = 3
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Solve y = log 121 ( 11 ) without using a calculator.

log 121 ( 11 ) = 1 2 (recalling that 121 = ( 121 ) 1 2 = 11 )

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Evaluating the logarithm of a reciprocal

Evaluate y = log 3 ( 1 27 ) without using a calculator.

First we rewrite the logarithm in exponential form: 3 y = 1 27 . Next, we ask, “To what exponent must 3 be raised in order to get 1 27 ?

We know 3 3 = 27 , but what must we do to get the reciprocal, 1 27 ? Recall from working with exponents that b a = 1 b a . We use this information to write

3 3 = 1 3 3 = 1 27

Therefore, log 3 ( 1 27 ) = 3.

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Evaluate y = log 2 ( 1 32 ) without using a calculator.

log 2 ( 1 32 ) = 5

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Using common logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression log ( x ) means log 10 ( x ) . We call a base-10 logarithm a common logarithm . Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition of the common logarithm

A common logarithm    is a logarithm with base 10. We write log 10 ( x ) simply as log ( x ) . The common logarithm of a positive number x satisfies the following definition.

For x > 0 ,

y = log ( x )  is equivalent to  10 y = x

We read log ( x ) as, “the logarithm with base 10 of x ” or “log base 10 of x .

The logarithm y is the exponent to which 10 must be raised to get x .

Given a common logarithm of the form y = log ( x ) , evaluate it mentally.

  1. Rewrite the argument x as a power of 10 : 10 y = x .
  2. Use previous knowledge of powers of 10 to identify y by asking, “To what exponent must 10 be raised in order to get x ?

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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