Exponential equation

Definition

An exponential equation is an equation in which the variables appear as exponents. These equations often take the form $a^{x} = b$ where $a$ and $b$ are constants.

5 Must Know Facts For Your Next Test

1. Exponential equations can be solved by taking the logarithm of both sides.
2. If bases are the same on both sides of the equation, their exponents can be set equal to each other.
3. Natural logarithms (ln) are often used when solving exponential equations involving base $e$.
4. Graphing can help visualize solutions to some exponential equations, particularly those with multiple or complex solutions.
5. Changing the base using logarithmic properties can simplify solving certain exponential equations.

Review Questions

• How do you solve an exponential equation where the base is not easily converted?
• What property allows you to equate exponents when bases are identical on both sides of an exponential equation?
• Which logarithm is typically used when dealing with base $e$ in an exponential equation?

Related terms

Logarithmic Equation: An equation that involves a logarithm with a variable argument, typically taking forms like $\log_b(x) = y$.

Base: The constant value that is raised to a power in an exponential expression, such as $a$ in $a^x$.

$e$ (Euler's Number): $e$ is an irrational and transcendental number approximately equal to 2.71828, commonly used as the base for natural logarithms.