"I'm thinking of another number," you tell your best friend. "The number I'm thinking of satisfies the equation
Guidance
A logarithmic equation has the variable within the log. To solve a logarithmic equation, you will need to use the inverse property,
Example A
Solve
Solution: There are two different ways to solve this equation. The first is to use the definition of a logarithm.
The second way to solve this equation is to put everything into the exponent of a 2, and then use the inverse property.
Make sure to check your answers for logarithmic equations. There can be times when you get an extraneous solution.
Example B
Solve
Solution: First, add 5 to both sides and then divide by 3 to isolate the natural log.
Recall that the inverse of the natural log is the natural number. Therefore, everything needs to be put into the exponent of
Checking the answer, we have
Example C
Solve
Solution: Condense the lefthand side using the Product Property.
Now, put everything in the exponent of 10 and solve for
Now, check both answers.
4 is an extraneous solution. In the step
Intro Problem Revisit
We can rewrite
Therefore, the number you are thinking of is 100.
Guided Practice
Solve the following logarithmic equations.
1.
2.
3.
Answers
1. Isolate the log and put everything in the exponent of 3.
2. Condense the lefthand side using the Quotient Rule and put everything in the exponent of
Checking our answer, we get
3. Multiply both sides by 2 and put everything in the exponent of a 5.
Explore More
Use properties of logarithms and a calculator to solve the following equations for

log2x=15 
log12x=2.5 
log9(x−5)=2 
log7(2x+3)=3 
8ln(3−x)=5 
4log33x−log3x=5 
log(x+5)+logx=log14 
2lnx−lnx=0 
3log3(x−5)=3 
23log3x=2 
5logx2−3log1x=log8 
2lnxe+2−lnx=10 
2log6x+1=log6(5x+4) 
2log12x+2=log12(x+10) 
3log23x−log2327=log238