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
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.
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
If bL = x
logbx = L
Rewrite in logarithmic form:
a. 5x = 625
b. 34 = x
c. b2 = 100
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
Rewrite in exponential form:
a. log6x = 2
b. log464 = x
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