Solving Exponential and Logarithmic Equations

There are a few strategies for solving exponential and logarithmic equations:

1. Rewrite the given equation in a form to use the One-to-One Properties of exponential or logarithmic functions. 
2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 
3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of logarithmic functions. 

One-to-One Properties: 

 if and only if 

and 

if and only if 

Inverse Properties:

Logarithmic equations can also be manipulated by exponentiation.

Notice that: 

•  is not a solution as it is extraneous 
• Be sure to check for extraneous solutions
• You can also check for extraneous solutions by graphing 

• Exponentiation is executed after the logarithmic expression has been isolated 

Investment: 

It's important to remember the formula for continually compounded interest: 

where A is the final balance, P is the principal, r is the rate and t is the time

Tips: 

• To solve exponential equations, it is useful to first isolate the exponential expression, then take the logarithm of both sides and solve for the variable. 
• To solve logarithmic equations, condense the logarithmic part into a single logarithm, then rewrite in exponential form and solve for the variable.