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Finding the value of a common logarithm mentally

Evaluate y = log ( 1000 ) without using a calculator.

First we rewrite the logarithm in exponential form: 10 y = 1000. Next, we ask, “To what exponent must 10 be raised in order to get 1000?” We know

10 3 = 1000

Therefore, log ( 1000 ) = 3.

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Evaluate y = log ( 1, 000, 000 ) .

log ( 1 , 000 , 000 ) = 6

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Given a common logarithm with the form y = log ( x ) , evaluate it using a calculator.

  1. Press [LOG] .
  2. Enter the value given for x , followed by [ ) ] .
  3. Press [ENTER] .

Finding the value of a common logarithm using a calculator

Evaluate y = log ( 321 ) to four decimal places using a calculator.

  • Press [LOG] .
  • Enter 321 , followed by [ ) ] .
  • Press [ENTER] .

Rounding to four decimal places, log ( 321 ) 2.5065.

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Evaluate y = log ( 123 ) to four decimal places using a calculator.

log ( 123 ) 2.0899

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Rewriting and solving a real-world exponential model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation 10 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?

We begin by rewriting the exponential equation in logarithmic form.

10 x = 500 log ( 500 ) = x Use the definition of the common log .

Next we evaluate the logarithm using a calculator:

  • Press [LOG] .
  • Enter 500 , followed by [ ) ] .
  • Press [ENTER] .
  • To the nearest thousandth, log ( 500 ) 2.699.

The difference in magnitudes was about 2.699.

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The amount of energy released from one earthquake was 8,500 times greater than the amount of energy released from another. The equation 10 x = 8500 represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

The difference in magnitudes was about 3.929.

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Using 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, log e ( x ) , has its own notation, ln ( x ) .

Most values of ln ( x ) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, ln 1 = 0. For other natural logarithms, we can use the 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.

Definition of the natural logarithm

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

For x > 0 ,

y = ln ( x )  is equivalent to  e y = x

We read ln ( x ) 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 = e x and y = ln ( x ) are inverse functions, ln ( e x ) = x for all x and e = ln ( x ) x for x > 0.

Given a natural logarithm with the form y = ln ( x ) , evaluate it using a calculator.

  1. Press [LN] .
  2. Enter the value given for x , followed by [ ) ] .
  3. Press [ENTER] .

Questions & Answers

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
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The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
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A mathematical relation such that every input has only one out.
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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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