# Algebra II : Adding and Subtracting Logarithms

## Example Questions

### Example Question #31 : Adding And Subtracting Logarithms

Simplify the log:

Explanation:

This question will require the product and quotient properties.

Product:

Quotient:

Evaluate the first two terms according to the formula.

### Example Question #32 : Adding And Subtracting Logarithms

Combine all logs into a single log:

Explanation:

Evaluate the first term.  It can be rewritten by moving the coefficient as a power of the integer inside the log.

According to the product property of logs, when logs are added, the values inside the log can be multiplied together so that the log can be combined as one unit.

Subtract this log with .

According to the division property of logs, when two log values are subtracted, we can combine the logs and divide the numbers.

### Example Question #33 : Adding And Subtracting Logarithms

Explanation:

The log has a default base of 10.  Recall that if:

Evaluate each term.

since .

since .

### Example Question #34 : Adding And Subtracting Logarithms

Combine as one log:

Explanation:

According to log properties, when there is a coefficient in front of a log, the coefficient can be transferred as the exponent of the inner value of log.

Convert the expression.

Simplify the terms.

When the logs of a same base are added, we can multiply the inner terms together to combine the log as one whole.

### Example Question #35 : Adding And Subtracting Logarithms

Explanation:

Combine the logs together using the log properties.

This is also equivalent to:

OR:

### Example Question #36 : Adding And Subtracting Logarithms

Combine as one log:

Explanation:

Whenever logs of the same bases are added, the terms inside the logs can be multiplied.

Likewise, when logs are subtracted, they are divided instead.

Evaluate the first two logs.

The expression becomes .

Divide the numbers inside.

### Example Question #37 : Adding And Subtracting Logarithms

Combine as one log, if possible:

Explanation:

According to the log property, we can combine the expression as one log

Be careful not to simplify  and subtract the quantity of the log and the integer.

### Example Question #38 : Adding And Subtracting Logarithms

Combine as one log:

Explanation:

According to log rules, the coefficients of the logs can be raised as powers for the inner quantity of the log.  Rewrite the terms.

Subtract the .

### Example Question #39 : Adding And Subtracting Logarithms

Combine as one log:

Explanation:

According to the log property, we can bring coefficients of logs as powers of the inner term of the log.

Rewrite the expression.

### Example Question #40 : Adding And Subtracting Logarithms

Explanation:

According to log properties, when there is a coefficient in front of the log, the coefficient can be used as the exponent for the inner quantity of the log.

Rewrite the expressions.

The expression becomes:

When logs are added, they can be combined into one log by multiplying the two quantities.

When logs are subtracted, the quantities are then divided.

Reduce the fraction.