Math

Solving Problems Involving Logarithmic Functions

 

A logarithmic equation is an equation that involves at least one logarithmic function. In general, there are two cases for logarithmic equations.

 

Case 1) In this case, we have logarithmic functions with similar bases on both sides of an equation. If the bases are the same, then the arguments are equal:

 

Case 2) In this case, there is a single logarithmic function on one side of the equation. To solve this type of equations, one has to convert the logarithm to an exponential function and solve for the variable:

 

Example: Solve 

 

Step 1) We simplify the left-hand-side using the logarithmic properties:

Step 2) The equation that we obtained in step 1) has logarithmic functions with similar bases on both sides, therefore the arguments are equal:

 

Example: Solve

First, we need to convert the logarithmic function to an exponential:

we now evaluate the exponent on the left-hand-side and solve for x:

 

 

Example: Solve 

 

Step 1) We combine the two logarithms on the left-hand-side:

Step 2) Since the logarithms on both sides have the same base, we set the arguments equal to each other and solve for x:

 

Step 3) We can factor the quadratic equation in step 2) and solve for x:

 

Example: Solve 

 

Step 1) We factor the first logarithm and write the equation with a single logarithm:

Step 2) The zero on the right-hand-side of the equation can be written as:

Step 3) We put the equations from step1) and step 2) together: