# Algebra II : Log-Base-10

## Example Questions

### Example Question #1 : Log Base 10

Based on the definition of logarithms, what is  ?

100

2

4

3

10

3

Explanation:

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

### Example Question #2 : Log Base 10

Evaluate .

Explanation:

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

### Example Question #2 : Log Base 10

What is the value of ?

Explanation:

Base-10 logarithms are very easy if the operands are a power of .  Begin by rewriting the question:

Becomes...

because

Applying logarithm rules, you can factor out the :

Now,  is .

### Example Question #2 : Log Base 10

What is the value of ?

Round to the nearest hundreth.

Explanation:

Base-10 logarithms are very easy if the operands are a power of .  Begin by rewriting the question:

Becomes...

because

Applying logarithm rules, you can factor out the :

Now,  is .

### Example Question #3 : Log Base 10

Many textbooks use the following convention for logarithms:

What is the value of ?

Explanation:

Remember:

is the same as saying .

So when we ask "What is the value of ?", all we're asking is "10 raised to which power equals 1,000?" Or, in an expression:

.

From this, it should be easy to see that .

### Example Question #6 : Log Base 10

Evaluate the following expression:

Explanation:

Without a subscript a logarithmic expression is base 10.

The expression

The logarithmic expression is asking 10 raised to what power equals 1000 or what is x when

We know that

so

### Example Question #7 : Log Base 10

Assuming the value of  is positive, simplify:

Explanation:

Rewrite the logarithm in division.

As a log property, we can pull down the exponent of the power in front as the coefficient.

Cancel out the .

### Example Question #8 : Log Base 10

Solve the following:

Explanation:

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

For the second term,

### Example Question #4 : Log Base 10

Which of the following expressions is equivalent to the expression  ?

None of the other choices is correct.

Explanation:

By the reverse-FOIL method, we factor the polynomial as follows:

Therefore, we can use the property

as follows:

### Example Question #4 : Log Base 10

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: