We can solve an exponential equation by rewriting each side of the equation with the same base. Then, we set the exponents equal to each other and solve the unknown variable.

**Example 1: **Solve the exponential equation below.

$${9^{3x - 8}} = {1 \over {81}}$$

Notice that both \(9\) and \(81\) are powers of \(3\).

$$\eqalign{

9 &= {3^2} \cr

81 &= {3^4} \cr} $$

This transforms our original equation into

$${\left( {{3^2}} \right)^{3x - 8}} = {1 \over {{3^4}}}$$

Distribute the exponents on the left using the Power of a Power Property.

$${3^{6x - 16}} = {1 \over {{3^4}}}$$

Take the reciprocal of the fraction on the right side but make sure to switch the sign of the exponent from positive to negative.

$${3^{6x - 16}} = {3^{ - 4}}$$

If they are equal, then their exponents must be equal as well. This is where we set the exponents equal to each other then we solve for the variable \(x\).

$$\eqalign{

6x - 16 &= - 4 \cr

6x &= - 4 + 16 \cr

6x &= 12 \cr

x &= 2 \cr} $$

The final answer is \(\boxed{x=2}\). We can check it with the original equation to verify.

$$\eqalign{

{9^{3x - 8}} &= {1 \over {81}} \cr

{9^{3\left( {\color{red}2} \right) - 8}} &= {1 \over {81}} \cr

{9^{6 - 8}} &= {1 \over {81}} \cr

{9^{ - 2}} &= {1 \over {81}} \cr

{1 \over {{9^2}}} &= {1 \over {81}} \cr

{1 \over {81}} &= {1 \over {81}} \cr} $$

**Example 2: **Solve the exponential equation below.

$$\large{{2^{2x - 3}} \cdot {4^{x + 6}} = {{{8^{1 - 2x}}} \over {16}}}$$

**STEP 1:** Express all bases as powers of \(2\).

$${2^{2x - 3}} \cdot {\left( {{2^2}} \right)^{x + 6}} = {{{{\left( {{2^3}} \right)}^{1 - 2x}}} \over {{2^4}}}$$

**STEP 2:** Apply the Power of a Power Property of Exponent.

$${2^{2x - 3}} \cdot {2^{2x + 12}} = {{{2^{3 - 6x}}} \over {{2^4}}}$$

**STEP 3:** Apply the Product Property of Exponent on the left side. That means, copy the common base then add the exponents.

$${2^{4x + 9}} = {{{2^{3 - 6x}}} \over {{2^4}}}$$

**STEP 4:** Apply the Quotient Property of Exponent on the right side. That means, copy the common base then subtract the exponents.

$${2^{4x + 9}} = {2^{ - 6x - 1}}$$

**STEP 5:** Now, set the exponents equal to each other, then solve for the variable \(x\).

$$\eqalign{

4x + 9 &= - 6x - 1 \cr

4x + 6x &= - 1 - 9 \cr

10x &= - 10 \cr

x &= - 1 \cr} $$

The final answer is \(\boxed{x=-1}\).