# Precalculus : Express Logarithms in Expanded Form

## Example Questions

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### Example Question #1 : Express Logarithms In Expanded Form

Expand the following logarithmic expression:

Explanation:

We start expanding our logarithm by using the following property:

Now we have two terms, and we can further expand the first term with the following property:

Now we only have two logarithms left with nonlinear terms, which we can expand using one final property:

Using this property on our two terms with exponents, we obtain the final expanded expression:

### Example Question #1 : Express Logarithms In Expanded Form

Expand

.

Explanation:

To expand

, use the quotient property of logs.

The quotient property states:

Substituting in our given information we get:

### Example Question #3 : Express Logarithms In Expanded Form

Which of the following correctly expresses the following logarithm in expanded form?

Explanation:

Begin by recalling a few logarithm rules:

1) When adding logarithms of like base, multiply the inside.

2) When subtracting logarithms of like base, subtract the inside.

3) When multiplying a logarithm by some number, raise the inside to that power.

Keep these rules in mind as we work backward to solve this problem:

Using rule 2), we can get the following:

Next, use rule 1) on the first part to get:

Finally, use rule 3) on the second and third parts to get our final answer:

### Example Question #1 : Express Logarithms In Expanded Form

Express  in its expanded, simplified form.

Explanation:

Using the properties of logarithms, expand the logrithm one step at a time:

When expanding logarithms, division becomes subtration, multiplication becomes division, and exponents become coefficients.

.

### Example Question #1 : Express Logarithms In Expanded Form

Expand this logarithm:

Explanation:

Use the Quotient property of Logarithms to express on a single line:

Use the Product property of Logarithms to expand the two terms further:

Finally use the Power property of Logarithms to remove all exponents:

The expression is now fully expanded.

### Example Question #1 : Express Logarithms In Expanded Form

Expand the following logarithm:

Explanation:

Expand the following logarithm:

To expand this log, we need to keep in mind 3 rules:

1) When dividing within a , we need to subtract

2) When multiplying within a , we need to add

3) When raising to a power within a , we need to multiply by that number

These will make more sense once we start applying them.

First, let's use rule number 1

Next, rule 2 sounds good.

Finally, use rule 3 to finish up!

### Example Question #1 : Express Logarithms In Expanded Form

Completely expand this logarithm:

Explanation:

Quotient property:

Product property:

Power property:

### Example Question #8 : Express Logarithms In Expanded Form

Fully expand:

Explanation:

In order to expand the expression, use the log rules of multiplication and division.  Anytime a variable is multiplied, the log is added.  If the variable is being divided, subtract instead.

When there is a power to a variable when it is inside the log, it can be pulled down in front of the log as a coefficient.

### Example Question #9 : Express Logarithms In Expanded Form

Expand the following:

Explanation:

To solve, simply remember that when you add logs, you multiply their insides.

Thus,

### Example Question #10 : Express Logarithms In Expanded Form

Express the following in expanded form.