Algebra Tutorials!
Friday 29th of September
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Logarithmic Functions

The statement 8 = 23 is in exponential form with base 2.

An equivalent way of writing 23 = 8 is the logarithmic form:

We read log28 = 3 as, â€œthe logarithm to the base 2 of 8 is 3.â€ Or, more briefly, â€œthe log base 2 of 8 is 3.â€

The statements 8 = 23 and log28 = 3 are two ways of expressing the same relationship between 8 and the cube of 2.

So what is a logarithm? A logarithm is an exponent. We can see this by noting the following: In the equation 8 = 23 we know 3 is the exponent.

If we rewrite 8 = 23 in logarithmic form we have log28 = 3. So the logarithm, log28, is equivalent to 3, an exponent.

Definition â€” Logarithmic Function

A logarithmic function is a function that has the form: f(x) =  logbx where b is a real number, b > 0, and b 1; x is a real number, x > 0.

The domain is all positive real numbers.

The range is all real numbers.

The inverse of an exponential function is a logarithmic function. This means that

â€¢ If f(x) = bx then f-1(x) = logbx. Both functions have base b.

â€¢ The domain of one function is the range of the other.

â€¢ Their graphs are mirror images of each other about the line y = x.

Since they are inverses, we can write an exponential equation as a log equation and vice versa.

Note:

The inverse of f(x) = 2x is f-1(x) = log2x.

The inverse of f(x) = log2x is f-1(x) = 2x.

Definition â€” Exponential and Logarithmic Forms

 Exponential formIf bL = x then Logarithmic formlogbx = L

Example 1

Rewrite in logarithmic form:

a. 5x = 625

b. 34 = x

c. b2 = 100

Solution

The statements are given in exponential form, bL = x.

Rewrite them in the equivalent logarithmic form, logbx = L.

 a. The base is 5. The exponent is x. b. The base is 3. The exponent is 4. c. The base is b. The exponent is 2. log5625 = x log3x = 4 logb100 = 2

Example 2

Rewrite in exponential form:

a. log6x = 2

b. log464 = x

c.

Solution

The statements are given in logarithmic form, logbx = L.

Rewrite them in the equivalent exponential form, bL = x.

 a. The base is 6. The exponent is 2. b. The base is 4. The exponent is x. c. The base is 5. The exponent is -2. 62 = x 4x = 64

Note:

Remember,