Skill: Solve exponential equations using common bases

To solve an exponential equation using common bases:

1. Rewrite both sides using the same positive base.
2. Use the one-to-one property: if $(0<a\ne 1)$ and $(a^u=a^v)$, then $(u=v)$.
3. Solve the resulting equation for the variable.
4. Check the answer if needed.

Example: Solve

$$4^{x+1}=8^{x-1}$$

Rewrite $4$ and $8$ as powers of the common base $2$:

$$4=2^2 \qquad\text{and}\qquad 8=2^3$$

Substitute these into the equation:

$$(2^2)^{x+1}=(2^3)^{x-1}$$

Use the power-of-a-power rule, $((a^m)^n=a^{mn})$:

$$2^{2(x+1)}=2^{3(x-1)}$$

Since both sides have the same base $2$, set the exponents equal:

$$2(x+1)=3(x-1)$$

Solve:

$$2x+2=3x-3$$ $$2=x-3$$ $$x=5$$

Therefore, the solution is

$$\boxed{x=5}$$

Check:

$$4^{5+1}=4^6=4096$$

and

$$8^{5-1}=8^4=4096$$

Both sides are equal, so $(x=5)$ is correct.