"I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation
Solving Logarithm Equations
A logarithmic equation has the variable within the log. To solve a logarithmic equation, you will need to use the inverse property,
Let's solve the following logarithmic equations.
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.
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
Condense the left-hand 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
Earlier, you were asked to determine what number you are thinking of if the number satisfies the equation
We can rewrite
Therefore, the number you are thinking of is 100.
Isolate the log and put everything in the exponent of 3.
Condense the left-hand side using the Quotient Rule and put everything in the exponent of
Checking our answer, we get
Multiply both sides by 2 and put everything in the exponent of a 5.
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
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 8.11.