Skill: Solve exponential equations using logarithms

To solve an exponential equation, use a logarithm to bring the exponent down where you can isolate it.

Consider the example

$$3^{2x-1}=20.$$

1. Take the logarithm of both sides.
   We can use either common logarithms or natural logarithms:

$$\log\left(3^{2x-1}\right)=\log(20).$$

2. Use the power property of logarithms, $\log(a^r)=r\log(a)$:

$$(2x-1)\log(3)=\log(20).$$

3. Divide by $\log(3)$ to isolate the exponent expression:

$$2x-1=\frac{\log(20)}{\log(3)}.$$

4. Solve for $x$:

$$2x=1+\frac{\log(20)}{\log(3)}$$ $$\boxed{x=\frac{1+\frac{\log(20)}{\log(3)}}{2}}.$$

Using a calculator,

$$\boxed{x\approx 1.863}.$$

5. Check the solution in the original equation:

$$3^{2(1.863)-1}\approx 3^{2.726}\approx 20.$$

The solution is reasonable because it makes the original equation true.

For an equation such as $a^{kx+c}=b$, the general process is

$$a^{kx+c}=b \quad\Longrightarrow\quad \log(a^{kx+c})=\log(b)$$ $$(kx+c)\log(a)=\log(b),$$

then solve the resulting linear equation for $x$. The base must satisfy $a>0$ and $a\ne1$, and the logarithm’s argument must be positive.