# Algebra II : Solving and Graphing Logarithms

## Example Questions

### Example Question #51 : Solving Logarithms

Solve .

Explanation:

The first thing we can do is write the coefficients in front of the logs as exponents of the terms inside the logs:

Next, we combine the logs:

Now we can change the equation into exponent form:

When we simplify, we'll also move all the terms to one side of the equation:

From here, we factor to get our solutions:

The last thing we have to do is check our answers.   doesn't raise any problems, but  does.  If we plug it back into the original equation, we would be evaluating a negative value in a log, which we can't do.

### Example Question #52 : Solving Logarithms

Solve:

Explanation:

In order to solve for the logs, we will need to write the log properties as follows:

and

This means that:

Replace the values into the expression.

### Example Question #53 : Solving Logarithms

Solve

Explanation:

First, we rewrite the equation as:

We can use log properties to simplify:

From here, simple algebra is used to solve:

### Example Question #54 : Solving Logarithms

Solve .

Explanation:

First we divide both sides of the equation by :

Next, we write the equation in exponential form:

Then we solve for  by taking the cube root, and then squaring :

### Example Question #55 : Solving Logarithms

Solve .

Explanation:

First we can divide each side by :

Next, we rewrite in exponent form:

From here, we simply take the square root of , and then subtract :

### Example Question #56 : Solving Logarithms

Solve

Explanation:

First we rewrite the equation in exponential form:

Now we take the cube root of :

### Example Question #57 : Solving Logarithms

Solve .

Explanation:

First we subtract  from both sides:

Then we divide both sides by :

Now it would help if we wrote the equation in exponential form (remember, if the log doesn't show a base, it's base 10):

Finally, we use algebra to solve:

### Example Question #201 : Logarithms

Solve

Explanation:

First, we can combine the log terms:

Now we can change to exponent form (remember, if a log doesn't specifically have a base, then it's base 10):

We need to set the equation equal to  in order to solve the quadratic equation, so we combine the  terms and subtract :

Then we factor and solve for :

Lastly, we have to check our answers.  When we plug  in to the original equation, everything comes out well.  However, when we use  we get errors (because you can't take the log of a negative number). Therefore, we only have 1 solution: .

### Example Question #202 : Logarithms

Solve .

Explanation:

First, we combine the logs:

Next, we can rewrite the equation in exponent form:

Now we use algebra to set the equation equal to :

We then factor and solve for :

Lastly, we have to check our answers.  Using  in the original equation results in errors (because you can't take the log of a negative number).   works out fine though, so we have 1 solution.

Solve