# Algebra II : Solving and Graphing Logarithms

## Example Questions

### Example Question #41 : Solving Logarithms

Find  if .

Explanation:

Now we can expand the right side of the equation:

A log of it's own base equals :

Now we add the logs on the right side of the equation by multiplying the terms inside the logs:

### Example Question #42 : Solving Logarithms

Solve

Explanation:

By definition, a logarithm of any base that has the term  inside is equal to .  So we set that term equal to :

### Example Question #43 : Solving Logarithms

Solve .

Explanation:

When a logarithm equals , the equation in the logarithm equals the logarithms base:

### Example Question #44 : Solving Logarithms

Solve

Explanation:

Rearranging the logarithm so that we exchange an exponent for the log we get:

### Example Question #45 : Solving Logarithms

Solve .

Explanation:

First we rearrange the equation, trading the logarithm for an exponent:

And then we solve:

### Example Question #46 : Solving Logarithms

Solve .

Explanation:

The first thing we can do is combine the log terms:

Now we can change to exponent form:

We can combine terms and set the equation equal to  to have a quadratic equation:

We then solve the equation and get the answers:

and

can't be an answer, because the values inside a log can't be negative, so that leaves us with a single answer of .

### Example Question #47 : Solving Logarithms

Solve .

Explanation:

The first thing we can do is combine log terms:

Simplifying the log term gives:

Now we can change the equation to exponent form:

And to solve:

Here, the solution can't be  because the term inside a logarithm can't be negative, so the only solution is .

### Example Question #48 : Solving Logarithms

Solve

Explanation:

First, let's change the equation to exponent form:

Then simplify:

And solve:

Both answers are valid because  in the original equation is squared, so any negative numbers don't cause the logarithm to become negative.

### Example Question #49 : Solving Logarithms

Solve

Explanation:

We can start by getting both the log terms on the same side of the equation:

Then we combine log terms:

Now we can change to exponent form:

Anything raised to the th power equals , and from here it becomes a simpler problem to solve:

### Example Question #50 : Solving Logarithms

Solve

Explanation:

First we're going to get all the natural logs on one side of the equation:

Next, we're going to combine all the terms into one natural log:

Now we can change to exponent form:

Anything raised to the th power equals , which helps us simplify:

From here, we can factor and solve:

We have to notice, however, that  isn't a valid answer because if we were to plug it into the original formula we would have a negative value in a logarithm.