Base

Definition

The base in an exponential function is the constant value that is raised to a variable exponent. In logarithmic functions, the base is the constant value that the logarithm operates on.

5 Must Know Facts For Your Next Test

1. In the exponential function $f(x) = b^x$, $b$ is the base.
2. The natural exponential function has a base of $e \approx 2.718$.
3. Logarithms are inverses of exponential functions, so if $y = b^x$, then $\log_b(y) = x$.
4. Changing the base in a logarithm can be done using the change of base formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ for any positive number $c$.
5. Common bases used in calculus are $10$, known as common logarithms, and $e$, known as natural logarithms.

Review Questions

• What is the base in the exponential function $f(x) = 3^x$?
• How do you express $\log_2(8)$ using a different base, such as $10$?
• If you have an exponential equation $y = e^x$, what would be its corresponding logarithmic form?

Related terms

Exponential Function:

A function of the form $f(x) = b^x$, where $b > 0$ and $b \neq 1$. The variable exponent makes it grow rapidly.

$e$ (Euler's Number): $e \approx 2.718$, it is an irrational number that is the base of natural logarithms and appears frequently in calculus.

$\log_b(x)$ (Logarithm): The inverse operation to exponentiation, meaning $\log_b(x) = y$ if and only if $b^y = x$. It determines how many times one must multiply the base by itself to obtain another number.