Solving Exponential Equations

Strategy 1: Same Base

If you can write both sides with the same base, set exponents equal.

Example: $2^x = 8$ $2^x = 2^3$ $x = 3$

Strategy 2: Take Logarithms

When bases can't match, use logarithms:

Example: $3^x = 7$ $\ln(3^x) = \ln(7)$ $x \ln(3) = \ln(7)$ $x = \frac{\ln(7)}{\ln(3)}$

Properties Used

Power Property: $\log(a^b) = b \log(a)$

One-to-One Property: If $b^x = b^y$, then $x = y$

Common Equations

Form: $a \cdot b^{cx} = d$

Steps:

1. Isolate the exponential term
2. Take log of both sides
3. Use power property
4. Solve for $x$

Example: $5 \cdot 2^{3x} = 40$ $2^{3x} = 8$ $2^{3x} = 2^3$ $3x = 3$ $x = 1$