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We can express the relationship between logarithmic form and its corresponding exponential form as follows:

log b ( x ) = y b y = x , b > 0 , b 1

Note that the base b is always positive.

Because logarithm is a function, it is most correctly written as log b ( x ) , using parentheses to denote function evaluation, just as we would with f ( x ) . However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as log b x . Note that many calculators require parentheses around the x .

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y = log b ( x ) and y = b x are inverse functions.

Definition of the logarithmic function

A logarithm    base b of a positive number x satisfies the following definition.

For x > 0 , b > 0 , b 1 ,

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

where,

  • we read log b ( x ) as, “the logarithm with base b of x ” or the “log base b of x . "
  • the logarithm y is the exponent to which b must be raised to get x .

Also, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b   is   ( 0 , ) .
  • the range of the logarithm function with base b   is   ( , ) .

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

Given an equation in logarithmic form log b ( x ) = y , convert it to exponential form.

  1. Examine the equation y = log b x and identify b , y , and x .
  2. Rewrite log b x = y as b y = x .

Converting from logarithmic form to exponential form

Write the following logarithmic equations in exponential form.

  1. log 6 ( 6 ) = 1 2
  2. log 3 ( 9 ) = 2

First, identify the values of b , y , and x . Then, write the equation in the form b y = x .

  1. log 6 ( 6 ) = 1 2

    Here, b = 6 , y = 1 2 , and   x = 6. Therefore, the equation log 6 ( 6 ) = 1 2 is equivalent to 6 1 2 = 6 .

  2. log 3 ( 9 ) = 2

    Here, b = 3 , y = 2 , and   x = 9. Therefore, the equation log 3 ( 9 ) = 2 is equivalent to 3 2 = 9.

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Write the following logarithmic equations in exponential form.

  1. log 10 ( 1, 000, 000 ) = 6
  2. log 5 ( 25 ) = 2
  1. log 10 ( 1 , 000 , 000 ) = 6 is equivalent to 10 6 = 1 , 000 , 000
  2. log 5 ( 25 ) = 2 is equivalent to 5 2 = 25
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Converting from exponential to logarithmic form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b , exponent x , and output y . Then we write x = log b ( y ) .

Converting from exponential form to logarithmic form

Write the following exponential equations in logarithmic form.

  1. 2 3 = 8
  2. 5 2 = 25
  3. 10 4 = 1 10,000

First, identify the values of b , y , and x . Then, write the equation in the form x = log b ( y ) .

  1. 2 3 = 8

    Here, b = 2 , x = 3 , and y = 8. Therefore, the equation 2 3 = 8 is equivalent to log 2 ( 8 ) = 3.

  2. 5 2 = 25

    Here, b = 5 , x = 2 , and y = 25. Therefore, the equation 5 2 = 25 is equivalent to log 5 ( 25 ) = 2.

  3. 10 4 = 1 10,000

    Here, b = 10 , x = 4 , and y = 1 10,000 . Therefore, the equation 10 4 = 1 10,000 is equivalent to log 10 ( 1 10,000 ) = 4.

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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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