Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
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
In the exponential function $f(x) = b^x$, $b$ is the base.
The natural exponential function has a base of $e \approx 2.718$.
Logarithms are inverses of exponential functions, so if $y = b^x$, then $\log_b(y) = x$.
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 $c$.
Common bases used in calculus are $10$, known as common logarithms, and $e$, known as natural logarithms.
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.