Skill: Use logarithms to solve exponential equations

To solve an exponential equation, first isolate the exponential expression. Then take a logarithm of both sides. Since logarithms and exponentials are inverse operations, the exponent can be brought down using the power property:

$$\log(a^r)=r\log(a).$$

Example: Solve

$$5\cdot 3^{2x-1}=40.$$

1. Isolate the exponential expression.
Divide both sides by $5$:

$$3^{2x-1}=8.$$

2. Take a logarithm of both sides.
You may use common logarithms or natural logarithms:

$$\log\left(3^{2x-1}\right)=\log(8).$$

3. Bring the exponent down.

$$(2x-1)\log(3)=\log(8).$$

4. Solve for $x$.
Divide by $\log(3)$:

$$2x-1=\frac{\log(8)}{\log(3)}.$$

Add $1$ and divide by $2$:

$$x=\frac{1+\frac{\log(8)}{\log(3)}}{2}.$$

This is an exact logarithmic answer. Using a calculator,

$$x\approx \frac{1+1.8928}{2}\approx 1.446.$$

So the solution is

$$\boxed{x\approx 1.446}.$$

Remember that logarithm arguments must be positive, so you can only take $\log$ of a positive quantity. Also, logarithms do not distribute over sums: $\log(a+b)\ne\log(a)+\log(b)$.