Lesson 6: Solving Exponential and Logarithmic Equations

Property

An exponential equation is one in which the variable is part of an exponent. Many exponential equations can be solved by writing both sides of the equation as powers with the same base. In general, if two equivalent powers have the same base, then their exponents must be equal. If $b^m = b^n$, then $m=n$ (for $b > 0, b \neq 1$).

Examples

• To solve the equation $3^x = 81$, rewrite it as $3^x = 3^4$. Equating the exponents gives the solution $x=4$.

• For the equation $5^{x-1} = 125$, write it as $5^{x-1} = 5^3$. Then, solve the simpler equation $x-1 = 3$ to get $x=4$.

Property

When it is not possible to write the expressions with the same base, take the common logarithm or natural logarithm of both sides once the exponential is isolated. Then use the Power Property to move the exponent to the front as a factor and solve for the variable.

Examples

• Solve $3^x = 20$. Take the common log of both sides: $\log(3^x) = \log(20)$. Using the Power Property, $x \log 3 = \log 20$. So, $x = \frac{\log 20}{\log 3} \approx 2.727$.

• Solve $4e^{x-1} = 30$. First, isolate the exponential: $e^{x-1} = 7.5$. Take the natural log of both sides: $\ln(e^{x-1}) = \ln(7.5)$. This simplifies to $x-1 = \ln(7.5)$, so $x = \ln(7.5) + 1 \approx 3.015$.