Common base

Definition

A common base is a shared base for exponential expressions that allows the equations to be manipulated and solved more easily. In solving exponential equations, converting terms to the same base simplifies the comparison of their exponents.

5 Must Know Facts For Your Next Test

1. To solve exponential equations with a common base, rewrite each term so they share the same base before setting their exponents equal.
2. Common bases are often simple integers like 2, 3, or 10.
3. If $a^x = a^y$ and $a > 0$, then $x = y$.
4. The ability to recognize and convert different bases to a common one is essential in simplifying complex equations.
5. Logarithms can also be used when it is difficult to identify a common base directly.

Review Questions

• How do you solve an equation where both sides have the same base?
• What steps would you take if an equation's terms do not initially have a common base?
• Why is it useful to rewrite exponential terms with a common base?

Related terms

Exponential Equation: An equation in which variables appear as exponents.

Logarithm: The inverse operation to exponentiation, indicating what power a number (the base) must be raised to produce another number.

$\log_b(a)$: $\log_b(a)$ represents the logarithm of $a$ with base $b$, answering the question: 'to what power must $b$ be raised, to obtain $a$?'