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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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Questions & Answers

what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
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.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
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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