# Precalculus : Properties of Logarithms

## Example Questions

### Example Question #11 : Properties Of Logarithms

Condense the following expression into one logarithm:

Explanation:

To condense this, the second term must have the 2 in front moved inside.

When adding two logs, multiply their insides; when subtracting two logs, divide their insides.

### Example Question #12 : Properties Of Logarithms

Expand the following log completely

Explanation:

To expand a logarithm, quantities in the inside that are multiplied get added and quantities in the inside that are divided get subtracted.

### Example Question #21 : Exponential And Logarithmic Functions

Solve for :

Explanation:

The first step to solving this problem is to realize that

Then, the equation falls into the follow form which resembles a quadratic.

Let . Then,

Thus, and .

Since ,

### Example Question #14 : Properties Of Logarithms

Simplify the expression.

Explanation:

Using the quotient rule for logarithms we can condense these two logarithms into a single logarithm.

We then obtain our answer by simple division.

### Example Question #15 : Properties Of Logarithms

Simplify the expression.

Explanation:

Using the properties of logarithms we first simplify the expression to .  Then we use the quotient rule for logarithms and cancel some terms to obtain our answer.

### Example Question #16 : Properties Of Logarithms

Simplify the expression.

Explanation:

Using the properties of logarithms we first rewrite the expression as . Now we combine the three pieces into the form . We then obtain our answer when we combine the terms with and cancel those with .

Solve for x:

Explanation:

### Example Question #21 : Exponential And Logarithmic Functions

Which of the following is equivalent to  ?

Explanation:

When multiplying exponents with a common base, you add both the exponents together. Hence,

When an exponent is raised to an exponent you multiply the exponents together. Hence, for  you would multiply  to get .

### Example Question #19 : Properties Of Logarithms

Solve the following for x:

Explanation:

Solve for x: