Understanding Logarithms

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Algebra II › Understanding Logarithms

Questions 1 - 10
1

Solve the following:

Explanation

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

is asking:

For the second term,

is asking:

So, our final answer is

2

Solve:

Explanation

Change the base of the inner term or log to base ten.

According to the log property:

The log based ten and the ten to the power of will cancel, leaving just the power.

The answer is:

3

Solve:

Explanation

Change the base of the inner term or log to base ten.

According to the log property:

The log based ten and the ten to the power of will cancel, leaving just the power.

The answer is:

4

Solve the following:

Explanation

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

is asking:

For the second term,

is asking:

So, our final answer is

5

Determine the value of:

Explanation

The natural log has a default base of .

According to the rule of logs, we can use:

The coefficient in front of the natural log can be transferred as the power of the exponent.

The natural log and base e will cancel, leaving just the exponent.

The answer is:

6

Given the following:

Decide if the following expression is true or false:

for all positive .

True

False

Explanation

By definition of a logarithm,

if and only if

Take the th root of both sides, or, equivalently, raise both sides to the power of , and apply the Power of a Power Property:

or

By definition, it follows that , so the statement is true.

7

, with positive and not equal to 1.

Which of the following is true of for all such ?

Explanation

By definition,

If and only if

Square both sides, and apply the Power of a Power Property to the left expression:

It follows that for all positive not equal to 1,

for all .

8

Evaluate to the nearest tenth.

Explanation

Since most calculators only have common and natural logarithm keys, this can best be solved as follows:

By the Change of Base Property of Logarithms, if and ,

Setting , we can restate this logarithm as the quotient of two common logarithms, and calculate accordingly:

or, when rounded, 2.5.

This can also be done with natural logarithms, yielding the same result.

Rond to one decimal place.

9

Evaluate .

Explanation

The first thing we can do is bring the exponent out of the log, to the front:

Next, we evaluate :

Recall that log without a specified base is base 10 thus

.

Therefore

becomes,

.

Finally, we do the simple multiplication:

10

Evaluate:

Explanation

We will need to write fraction in terms of the given base of log, which is ten.

According to the log rules:

This means that the expression of log based 10 and the power can be simplified.

The answer is:

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