# Algebra II : Adding and Subtracting Logarithms

## Example Questions

### Example Question #21 : Adding And Subtracting Logarithms

Explanation:

When adding logarithms of the same base, all you have to do is multiply the numbers inside the function as shown below:

### Example Question #22 : Adding And Subtracting Logarithms

Explanation:

When adding logarithms of the same base, all you have to do is multiply the numbers inside the function as shown below:

### Example Question #23 : Adding And Subtracting Logarithms

Simplify the following expression:

Explanation:

The furthest we can simplify the given expression is combining the two terms with the same base by multiplying the numbers on the "outside" of the logarithm (addition becomes multiplication when performing logarithmic operations).

### Example Question #24 : Adding And Subtracting Logarithms

If  and , what is ?

Explanation:

When adding logarithms, you multiply the terms inside the log functions.  For this problem you would have:

At this point, you FOIL (First, Outer, Inner, Last) the terms to get:

You then collect the like terms together:

Remember, these terms were originally in a log function, so to get our answer we need to put them back in one:

### Example Question #25 : Adding And Subtracting Logarithms

If  and , what is

Explanation:

The first thing we're going to do is to write the whole problem out so we know what we're dealing with:

The next thing is to move the  in front of the second term to the exponent of the term inside the log function:

Because we're subtracting the logs, we need to divide the terms inside of them:

Now we can cancel one of the  terms, leaving us with a final answer of:

### Example Question #26 : Adding And Subtracting Logarithms

If , and , what is ?

Explanation:

That doesn't specifically help us, but we can rewrite it in the form of:

From here, we can split it into two separate logarithms:

We know the values for these terms, so we can substitute them in and solve now:

### Example Question #27 : Adding And Subtracting Logarithms

If  and , what is ?

Explanation:

When you add logarithms of the same base, you multiply the terms inside the log functions.  With this problem, you would end up with:

To multiply these terms together, you multiply the constants out in front like normal, and you add the exponents:

Now we need to remember to put this term back in the logarithm to get our final answer:

### Example Question #28 : Adding And Subtracting Logarithms

Condense the expression

Explanation:

When simplifying logarithmic expressions it is important to recall the rules of logs.

When two logs of the same base are added together that is the same as multiplying the inside contents.

When two logs of the same base are being subtracted that is the same as dividing the inside contents.

When there is a coefficient on a log that is the same as raising the log to that power.

Applying the rule above to the particular expression in question results in the following.

### Example Question #29 : Adding And Subtracting Logarithms

Combine the logarithms as a single log:

Explanation:

Use the log rules to simplify this expression.

Evaluate the first two terms.

Evaluate   and combine this into one log.

### Example Question #30 : Adding And Subtracting Logarithms

Simplify the logarithms: