## Algebra## Exponents and Logarithms |

## How are equations with exponents and logarithms solved? |

The way to solve an exponential equation is relatively easy: Take the log of both sides of the equation, then solve for the variable. For example, to solve for *x* in the equation *e ^{x}* = 60.

- First, take the natural log (ln) of both sides:
ln(e

^{x}) = ln(60) - Simplify using the logarithmic rule #3 (see above) for the left side:
*x*ln*(e)*= ln(60) - Then simplify again, since ln(
*e*) = 1 to:*x=*ln(60) = 4.094344562 - And finally, check your answer (using log tables or your calculator) in the original equation
*e*= 60:^{x}*e*^{4.094344562}= 60 is definitely true.

The way to solve a logarithmic equation is equally easy: Just rewrite the equation in exponential form and solve for the variable. For example, to solve for *x* in the equation ln(*x*) = 11:

- 1.First, change both sides so they are exponents of the base e:
*e*=^{ln(x)}*e*^{11} - When the bases of the exponent and logarithm are the same, the left part of the equation becomes
*x,*thus, it can be written:*x*=*e*^{11} - To obtain
*x,*determine the solution for*e*^{11}, or*x*is approximately 59,874.14172. - And finally, check your answer (using tables or your calculator) in the original equation ln
*(x)*= 11:ln(59,874.14172) = 11 is definitely true.