Guided Example #7 - Solving Equations with Unknown Exponents

Solving Equations with unknown exponents

If an unknown value is the power of a term (e.g. $e^{x} or 10^{x}$ ) and its value is to be calculated, then we must take logs on both sides of the equation to allow it to be solved. The same solution will be reached using any base, but calculators can be used for evaluating logs to the base e and 10.

Example

Solve $e^{x} = 7$

Taking $\log_{e}$ $\log_e$ of both sides

$x\log_ee=\log_e7\left(\log_ee=1\right)x=\log_e7x\approx1.946$ $x\log_ee=\log_e7\left(\log_ee=1\right)x=\log_e7x\approx1.946$