How to solve an exponential equation with different bases

When the bases of an exponential equation are different, you cannot simply set the exponents equal. The two standard tactics are (1) rewrite each side with a common base, and (2) apply logarithms to both sides to bring the exponents down. Choosing the right tactic depends on whether the bases share a factor that is itself a power of an integer. Both approaches lead to a linear equation in the unknown exponent, which can then be solved with ordinary algebra. The first method often yields a cleaner result, but it only works when a common base exists. The second method works for any positive bases.

Method 1: Convert to a Common Base

Example using a common base: solve $8^{x}=2^{x+6}$. Notice that $8=2^{3}$, so rewrite the left side as $(2^{3})^{x}=2^{3x}$. The equation becomes $2^{3x}=2^{x+6}$. Because the bases are now identical, set the exponents equal: $3x = x + 6$. Subtract $x$ from both sides to obtain $2x = 6$ and thus $x = 3$. This technique only works when each original base can be expressed as a power of the same integer.

Method 2: Use Logarithms

Follow these steps to apply logarithms:

1. 1Verify both bases are positive and not equal to 1
2. 2Take the same logarithm (natural or base‑10) of both sides
3. 3Use the power rule $\log(a^{b}) = b\log a$ to bring the exponent down
4. 4Solve the resulting linear equation for the variable

Key logarithm properties you will use:

• $\log(ab)=\log a+\log b$
• $\log\left(\frac{a}{b}\right)=\log a-\log b$
• $\log(a^{c})=c\log a$
• $\log_{k}k=1$

Example using logarithms: solve $5^{2x}=12$. The bases 5 and 12 share no common power, so take the natural logarithm of both sides: $\ln(5^{2x}) = \ln 12$. Apply the power rule to get $2x\ln 5 = \ln 12$. Divide by $2\ln 5$: $x = \frac{\ln 12}{2\ln 5}$. Numerically, $\ln 12 \approx 2.4849$ and $\ln 5 \approx 1.6094$, so $x \approx 0.772$. The same result is obtained with base‑10 logs because the change‑of‑base factor cancels.

Comparison of the two methods: