Solving Exponential Equations

An exponential equation is an equation where the variable appears in the exponent. To solve these equations, we generally use one of two methods depending on the numbers involved: finding a common base or using logarithms.

Method 1: Finding a Common Base

If both sides of the equation can be written as powers of the same base, you can simply set the exponents equal to each other.

The Rule: If $b^x = b^y$, then $x = y$ (where $b > 0$ and $b \neq 1$).

Example: Solve $3^{2x - 1} = 27$

1. Recognize that $27$ can be written as a power of $3$: $27 = 3^3$.
2. Rewrite the equation with the common base: $3^{2x - 1} = 3^3$
3. Since the bases are the same, set the exponents equal: $2x - 1 = 3$
4. Solve for $x$: $2x = 4$ $x = 2$

Method 2: Taking Logarithms of Both Sides

When the bases on both sides cannot be easily matched, we use logarithms. You can take the common logarithm ($\log$) or the natural logarithm ($\ln$) of both sides.

The Rule: Use the power rule of logarithms, $\log(b^x) = x \log(b)$, to bring the variable down from the exponent.

Example: Solve $5^x = 12$

1. Since $12$ is not a neat power of $5$, take the natural logarithm ($\ln$) of both sides: $\ln(5^x) = \ln(12)$
2. Use the logarithm power rule to move the $x$ to the front: $x \ln(5) = \ln(12)$
3. Isolate $x$ to get the exact answer: $x = \frac{\ln(12)}{\ln(5)}$
4. Calculate the approximate answer using a calculator: $x \approx 1.544$

Summary

• Check for a common base first. It is usually the fastest method.
• Use logarithms if the bases don't match. Remember to apply the power rule to bring the exponent down so you can solve for the variable algebraically.