# Precalculus : Evaluate Logarithms

## Example Questions

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### Example Question #1 : Evaluate Logarithms

Find .

Explanation:

Using the Change of Base formula we can rewrite the log as follows.

.

Or,

, then find that .

Therefore,

.

### Example Question #2 : Evaluate Logarithms

Evaluate the following logarithm:

Explanation:

The problem asks us to evaluate the following logarithm:

According to the definition of a logarithm, what this equation is asking us is "5 to what power equals 7?":

Using the properties of logarithms, if we take the log of both sides we get the following equation and simplification:

### Example Question #3 : Evaluate Logarithms

Evaluate the following logarithmic expression:

Explanation:

In order to evaluate the logarithmic expression, we have to remember the notation of a logarithm and what it means:

Any logarithm expressed in the form above is simply asking "a to what power equals b?" So we're trying to find what power the base must be raised to in order to obtain the value in parentheses. If we look at our expression in particular:

The first term is asking "5 to what power equals 1/25?" while the second is asking "7 to what power equals 49?" Setting these questions up mathematically, we can find the values of each logarithm:

So the value of the first term is -2, and the value of the second term is 2, which gives us:

### Example Question #4 : Evaluate Logarithms

Evaluate the following logarithm:

Explanation:

The notation of this logarithm is asking "5 to what power equals 125?" If we set this up mathematically, we can simplify either side until it is readily apparent what the value of x is, which is the value of the logarithm:

So to answer our question, 5 to the power of 3 is 125, so the answer is 3.

### Example Question #5 : Evaluate Logarithms

Evaluate:

Explanation:

To evaluate, you must convert 256 to a power of 2. Then the log and 2 will cancel to leave you with your answer.

### Example Question #6 : Evaluate Logarithms

Which of these is a correct method for evaluating ?

Explanation:

When evaluating a log you are asking what power your base must be raised by to get your argument.

If you cannot directly figure that out you can use a change of base formula instead.

### Example Question #7 : Evaluate Logarithms

Evaluate this logarithm:

Explanation:

It is important to remember a logarithm is really just an exponent. In fact when you see  remember that the expression is only asking what exponent must "a" be raised to in order to obtain "x"!

Now that we have substituted the value of x reread the problem by the rule above. What power must 3 be raised to in order to obtain 243?

5 is the power to which 3 must be raised to obtain 243.

### Example Question #8 : Evaluate Logarithms

Evaluate the logarithm

Explanation:

Given the property

and because

we obtain that

### Example Question #9 : Evaluate Logarithms

Evaluate to four decimal places:

Explanation:

Using a calculator, evaluate

### Example Question #10 : Evaluate Logarithms

Evaluate to 4 decimal places: