Exponential and Logarithmic Equations

There are two basic strategies for solving logarithmic equations. These strategies are based on two properties.

1. One-to-One Properties:

a^x=a^y if and only if x=y

log_ax=log_ay if and only if x=y

2.Inverse Properties

a^log_ax=x

log_aa^x=x

When solving both exponential and logarithmic equations, it is usually necessary to take either the log or natural log of both sides of the equation, (applicable to exponential equations) or exponentiate both sides of the equation. (applicable for logarithmic equations)

Exponential equation example:

(taking the natural log of both sides as seen in step 3)

Logarithmic equation example:

(exponentiation is seen in the step 2)

Its possible to solve exponential and logarithmic equations in quadratic forms and these are to be solved in the same method as other quadratic equations by factoring or the quadratic formula. It is important to check for extraneous solutions at the end of solving these types of problems.