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Write the following exponential equations in logarithmic form.

  1. 32=9
  2. 53=125
  3. 21=12
  1. 32=9 is equivalent to log3(9)=2
  2. 53=125 is equivalent to log5(125)=3
  3. 21=12 is equivalent to log2(12)=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 log28. We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know 23=8, it follows that log28=3.

Now consider solving log749 and log327 mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know 72=49. Therefore, log749=2
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know 33=27. Therefore, log327=3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log2349 mentally.

  • We ask, “To what exponent must 23 be raised in order to get 49? ” We know 22=4 and 32=9, so (23)2=49. Therefore, log23(49)=2.

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

  1. Rewrite the argument x as a power of b: by=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=log4(64) without using a calculator.

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

We know

43=64

Therefore,

log(64)4=3
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Solve y=log121(11) without using a calculator.

log121(11)=12 (recalling that 121=(121)12=11 )

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Evaluating the logarithm of a reciprocal

Evaluate y=log3(127) without using a calculator.

First we rewrite the logarithm in exponential form: 3y=127. Next, we ask, “To what exponent must 3 be raised in order to get 127?

We know 33=27, but what must we do to get the reciprocal, 127? Recall from working with exponents that ba=1ba. We use this information to write

33=133=127

Therefore, log3(127)=3.

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Evaluate y=log2(132) without using a calculator.

log2(132)=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 log10(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 log10(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 10y=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: 10y=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?
Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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