Common base

Definition

A common base refers to the shared base in exponential and logarithmic equations that allows for simplification and solving. It is often used to compare, combine, or solve exponential expressions by rewriting them with the same base.

5 Must Know Facts For Your Next Test

1. When solving exponential equations, rewrite both sides of the equation with a common base.
2. Common bases frequently used include 2, e (Euler's number), and 10.
3. Logarithmic equations can often be simplified by converting terms to a common base.
4. Having a common base allows the use of properties like $a^m = a^n \Rightarrow m = n$ to solve equations.
5. In some cases, you may need to use logarithms to find the common base if it isn't immediately apparent.

Review Questions

• How do you simplify $2^{3x} = 8$ using a common base?
• What property allows you to set exponents equal when working with a common base?
• Explain how using a common base can simplify the process of solving logarithmic equations.

Related terms

Exponential Function:

A function of the form $f(x) = a \cdot b^x$ where $b > 0$ and $b \neq 1$.

Logarithm:

The inverse operation to exponentiation, denoted as $\log_b(x)$ meaning the power to which the base $b$ must be raised to yield $x$.

Change of Base Formula:

$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, which allows conversion between different logarithmic bases.