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Study Guides > Intermediate Algebra

Read: Evaluate Logarithms

Learning Objectives

  •  Mentally evaluate logarithms
  • Define natural logarithm, evaluate natural logarithms with a calculator
  • Define common logarithm, evaluate common logarithms mentally and with a calculator

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log28{\mathrm{log}}_{2}8. We ask, "To what exponent must 22 be raised in order to get 88?" Because we already know 23=8{2}^{3}=8, it follows that log28=3{\mathrm{log}}_{2}8=3.

Now consider solving log749{\mathrm{log}}_{7}49 and log327{\mathrm{log}}_{3}27 mentally.

  • We ask, "To what exponent must 77 be raised in order to get 4949?" We know 72=49{7}^{2}=49. Therefore, log749=2{\mathrm{log}}_{7}49=2
  • We ask, "To what exponent must 33 be raised in order to get 2727?" We know 33=27{3}^{3}=27. Therefore, log327=3{\mathrm{log}}_{3}27=3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log2349{\mathrm{log}}_{\frac{2}{3}}\frac{4}{9} mentally.

  • We ask, "To what exponent must 23\frac{2}{3} be raised in order to get 49\frac{4}{9}? " We know 22=4{2}^{2}=4 and 32=9{3}^{2}=9, so (23)2=49{\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}. Therefore, log23(49)=2{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2.
In our first example we will evaluate logarithms mentally.

Example

Solve y=log4(64)y={\mathrm{log}}_{4}\left(64\right) without using a calculator.

Answer:

First we rewrite the logarithm in exponential form: 4y=64{4}^{y}=64. Next, we ask, "To what exponent must 44 be raised in order to get 6464?"

We know
43=64{4}^{3}=64

Therefore,

log4(64)=3\mathrm{log}{}_{4}\left(64\right)=3

In our first video we will show more examples of evaluating logarithms mentally, this helps you get familiar with what a logarithm represents. https://youtu.be/dxj5J9OpWGA In our next example we will evaluate the logarithm of a reciprocal.

Example

Evaluate y=log3(127)y={\mathrm{log}}_{3}\left(\frac{1}{27}\right) without using a calculator.

Answer:

First we rewrite the logarithm in exponential form: 3y=127{3}^{y}=\frac{1}{27}. Next, we ask, "To what exponent must 33 be raised in order to get 127\frac{1}{27}"?

We know 33=27{3}^{3}=27, but what must we do to get the reciprocal, 127\frac{1}{27}? Recall from working with exponents that ba=1ba{b}^{-a}=\frac{1}{{b}^{a}}. We use this information to write

\begin{array}{3}^{-3}=\frac{1}{{3}^{3}}\hfill \\ =\frac{1}{27}\hfill \end{array}

Therefore, log3(127)=3{\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3.

 

How To: Given a logarithm of the form y=logb(x)y={\mathrm{log}}_{b}\left(x\right), evaluate it mentally.

  1. Rewrite the argument x as a power of b: by=x{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?"

 Natural logarithms

The most frequently used base for logarithms is e. Base e logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base e logarithm, loge(x){\mathrm{log}}_{e}\left(x\right), has its own notation, ln(x)\mathrm{ln}\left(x\right).

Most values of ln(x)\mathrm{ln}\left(x\right) can be found only using a calculator. The major exception is that, because the logarithm of 11 is always 00 in any base, ln1=0\mathrm{ln}1=0. For other natural logarithms, we can use the ln\mathrm{ln} key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e using the inverse property of logarithms.

A General Note: Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e. We write loge(x){\mathrm{log}}_{e}\left(x\right) simply as ln(x)\mathrm{ln}\left(x\right). The natural logarithm of a positive number x satisfies the following definition.

For x>0x>0,

y=ln(x) is equivalent to ey=xy=\mathrm{ln}\left(x\right)\text{ is equivalent to }{e}^{y}=x

We read ln(x)\mathrm{ln}\left(x\right) as, "the logarithm with base e of x" or "the natural logarithm of x."

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

Since the functions y=exy=e{}^{x} and y=ln(x)y=\mathrm{ln}\left(x\right) are inverse functions, ln(ex)=x\mathrm{ln}\left({e}^{x}\right)=x for all x and eln(x)=xe{}^{\mathrm{ln}\left(x\right)}=x for 00.

In the next example, we will evaluate a natural logarithm using a calculator.

Example

Evaluate y=ln(500)y=\mathrm{ln}\left(500\right) to four decimal places using a calculator.

Answer:

  • Press [LN].
  • Enter 500500, followed by [ ) ].
  • Press [ENTER].

Rounding to four decimal places, ln(500)6.2146\mathrm{ln}\left(500\right)\approx 6.2146

In our next video, we show more examples of how to evaluate natural logarithms using a calculator. https://youtu.be/Rpounu3epSc

Common logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 1010. In other words, the expression log{\mathrm{log}}_{} means log10{\mathrm{log}}_{10}  We call a base-1010 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 Common Logarithm: Log is an exponent

A common logarithm is a logarithm with base 1010.  We write log10(x){\mathrm{log}}_{10}(x)  simpliy as log(x){\mathrm{log}}_{}(x).  The common logarithm of a positive number, x, satisfies the following definition: For x>0x\gt0

y=log(x)y={\mathrm{log}}_{}(x) is equivalent to 10y=x10^y=x

We read log(x){\mathrm{log}}_{}(x) as " the logarithm with base 1010 of x" or "log base 1010 of x".

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

Example

Evaluate log(1000){\mathrm{log}}_{}(1000) without using a calculator.

Answer: We know 103=100010^3=1000, therefore log(1000)=3{\mathrm{log}}_{}(1000)=3

Example

Evaluate y=log(321)y={\mathrm{log}}_{}(321) to four decimal places using a calculator.

Answer:

  • Press [LOG].
  • Enter 321321, followed by [ ) ].
  • Press [ENTER].
Rounding to four decimal places, log(321)2.5065{\mathrm{log}}_{}(321)\approx2.5065

In our last example we will use a logarithm to find the difference in magnitude of two different earthquakes.

Example

The amount of energy released from one earthquake was 500500 times greater than the amount of energy released from another. The equation 10x=50010^x=500 represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Answer: We begin by rewriting the exponential equation in logarithmic form.

10x=50010^x=500

log(500)=x{\mathrm{log}}_{}(500)=x

Next we evaluate the logarithm using a calculator:

  • Press [LOG].
  • Enter 500500 followed by [ ) ].
  • Press [ENTER].
  • To the nearest thousandth, log(500)2.699{\mathrm{log}}_{}(500)\approx2.699log(500)2.699.log(500)2.699.">

Summary

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally because the logarithm is an exponent.  Logarithms most commonly sue base 10, and often use base e. Logarithms can also be evaluated with most kinds of calculator.
 

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