Log-Base-10 - Algebra 2

This question tests your understanding of log functions.

can be converted to the form .

In this problem, make sure to divide both sides by in order to put it in the above form, where . Remember .

Therefore,

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

Therefore, we can use the property

as follows:

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

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

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 .

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 .

Therefore, your answer is .

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 .

Therefore, your answer is .

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

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 .

The answer is:

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

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:

One of the properties of logs is the ability to cancel out terms based on the base of the log. Since the base of the log is 10 we can simplify the 100 to 10 squared.

The log base 10 and the 10 cancel out, leaving you with the value of the exponent, 2 as the answer.

In order to eliminate the log based ten, we will need to raise both sides as the exponents using the base of ten.

The equation becomes:

The ten and log based ten will cancel, leaving just the power on the left side. Change the negative exponent into a fraction on the right side.

Divide by two on both sides, which is similar to multiplying by a half on both sides.

Simplify both sides.

The answer is:

Break up using log rules. The log has a default base of ten.

The exponent can be brought down as the coefficient since the bases of the second term are common.

This means that:

The answer is:

In order to solve this, we will need to rewrite the base as , since log is by default base 10.

Rewrite the expression.

By log rules, the exponent can be pulled down as the coefficient.

The answer is:

The log term has a default base of 10. The 1000 will need to be rewritten as base 10.

Raise the coefficient of the log term as the power.

According to the log property:

The log based 10 and the 10 inside the quantity of the log will cancel, leaving just the power.

The answer is:

The log has a default of base ten. This means we should convert the 1000 to a common base 10.

Replace this value inside the log term.

Since the log base 10 and the ten to a certain power are existent, they will both cancel, leaving just the power itself.

The answer is:

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:

Rewrite the log such that it is in its simplest form. Break up the 500 with common factors.

This can be broken into addition of logs.

The answer is: