# Algebra II : Solving and Graphing Logarithms

## Example Questions

### Example Question #31 : Solving Logarithms

Expand the log:

Explanation:

Rewrite the logarithm using the quotient property.

Rewrite the log.

Use the power property to pull down the power.

Simplify the log terms.

### Example Question #32 : Solving Logarithms

Solve for :

None of these

Explanation:

Using definition of logarithms:

### Example Question #33 : Solving Logarithms

Solve for :

None of these

Explanation:

Using definition of logarithms

### Example Question #34 : Solving Logarithms

Solve the following equation:

Explanation:

Recall the definition of a logarithm. For a logarithm in any base

denotes

.

In other words, the value of a logarithm is simply an exponent, and it is defined at  whenever the base can be raised to a sufficient power to yield . Hence, the equation

may be rewritten as

.

Raising  to the th power yields , and so

.

### Example Question #35 : Solving Logarithms

Solve the following equation:

Explanation:

According to the rule for multiplying logarithms,  for any constant value . In other words, an exponent on the quantity inside a logarithm can be moved outside the logarithm as a multiplier, and vice versa. Hence,

.

Since the natural logarithm has  as its base, the definition of logarithms applies in this case:  whenever the base of  is equal to .

.

This implies that  and . However, substituting the solution  into the original equation yields the following expression:

which is not defined, since logarithms are undefined for any value of .

Hence, we eliminate the extraneous solution  and submit that the correct solution to this equation is .

### Example Question #36 : Solving Logarithms

Solve for :

Explanation:

Use the following interchangeable properties to solve for the unknown variable:

For the equation , this would be equivalent to:

Divide by two on both sides and convert the negative exponent to a fraction.

### Example Question #37 : Solving Logarithms

Solve the log:

Explanation:

In order to simplify this log, we will need to convert the nine to base three.

Rewrite the term inside the log.

Rewrite this as a fraction.

According to log rules, we can pull the power of the inner quantity inside the log out as the coefficient.

### Example Question #38 : Solving Logarithms

Evaluate:

Explanation:

The expression can be rewritten as a fraction.

The log based 100 can be rewritten as ten squared.

### Example Question #39 : Solving Logarithms

Solve the logs:

Explanation:

For logs that are added, the inner quantities can be multiplied with each other to be combined as a single log.

For logs that are subtracted, the inner quantities would each be divided instead.

Rewrite the expression:

### Example Question #40 : Solving Logarithms

Evaluate:

Explanation:

Recall that log has a base of ten.

The equation can be rewritten as: