<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
Practice Key Terms 3

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask