When an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. We can set the exponents equal to one another and solve for the unknown.

For example, to solve the exponential equation $2^{x} = 8$, we might note that $8$ can be written as $2^{3}$.

$2^x=8$

$2^x=2^3$

$x=3$

In general, we can summarize solving exponential equations whose terms all have the same base in this way:

For any algebraic expressions S and T, and any positive real number $b\ne 1$

Exponential Equations with Unlike Bases

Sometimes it is not possible to write both sides of an exponential equation as powers of the same base. Base $e$ is a very common base found in science, finance, and engineering applications. When we have an equation with a base $e$ on either side, we can use the natural logarithm to solve it. Earlier, we introduced a formula that models continuous growth/decay, $y=Ae^{kt}$. To solve this equation for $t$,

• Isolate the exponential expression.
• Use the natural logarithm function to write the exponential equation in logarithmic form.
• Solve for $t$.

Sometimes an equation may not have a solution. For example, to solve $e^{t}= -2$ we might convert to logarithmic form to obtain $t=\mathrm{ln} \ (-2)$. Entering this on your calculator produces an error message. We can’t take a logarithm of a negative number. The original expression, $e^{t}= -2$ is never true since $e^{t}>0$ for all real numbers $t$.