# High School Math : Logarithms

## Example Questions

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### Example Question #1 : Logarithms

Explanation:

Most of us don't know what the exponent would be if  and unfortunately there is no  on a graphing calculator -- only  (which stands for ).

Fortunately we can use the base change rule:

Plug in our given values.

### Example Question #1 : Log Base 10

Based on the definition of logarithms, what is  ?

10

4

2

3

100

3

Explanation:

For any equation , . Thus, we are trying to determine what power of 10 is 1000. , so our answer is 3.

### Example Question #1 : Logarithms

What is the value of  that satisfies the equation  ?

Explanation:

is equivalent to . In this case, you know the value of  (the argument of a logarithmic equation) and b (the answer to the logarithmic equation). You must find a solution for the base.

### Example Question #1 : Simplifying Logarithms

Simplify .

Explanation:

Using properties of logs we get:

### Example Question #2 : Simplifying Logarithms

Simplify the following expression:

Explanation:

Recall the log rule:

In this particular case,  and . Thus, our answer is .

### Example Question #3 : Simplifying Logarithms

Use the properties of logarithms to solve the following equation:

No real solutions

Explanation:

Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:

The logarithm can be converted to exponential form:

Factor the equation:

Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is .

### Example Question #1 : Simplifying Logarithms

Which of the following represents a simplified form of

Explanation:

The rule for the addition of logarithms is as follows:

As an application of this,

### Example Question #141 : Mathematical Relationships And Basic Graphs

Simplify the expression using logarithmic identities.

The expression cannot be simplified

Explanation:

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.

### Example Question #1 : Simplifying Logarithms

Evaluate by hand

Cannot be found by hand

Explanation:

Using the logarithm rules, exponents within logarithms can be removed and simply multiplied by the remaining logarithm.  This expression can be simplified as

Solve for