# Completely expand as a sum/difference of logs

• lolilovepie
In summary: Could you provide a simplified version of what you are trying to say?In summary, the logarithm of 2x^4(x-15)^3 is -log12(x^4-16), while the logarithm of 12sqrt(x^4-16) is 3log2x^4.

## Homework Statement

Completely expand as a sum/difference of logs
log [ ( 2x^4(x-15)^3) / (12 sqrt (x^4-16) ]

## The Attempt at a Solution

log 2x^4(x-15)^3 - log 12 sqrt (x^4-16)
3 log 2x^4(x-15) - 1/2log 12 (x^4-16)

Last edited:
lolilovepie said:

## Homework Statement

Completely expand as a sum/difference of logs
log [ ( 2x^4(x-15)^3) / (12 sqrt (x^4-16) ]
Some clarification, please. The numerator of what you wrote is
2x4 * (x - 15)3.

If that isn't what you meant, please use parentheses or brackets to make it clearer.
lolilovepie said:

## The Attempt at a Solution

log 2x^4(x-15)^3 - log 12 sqrt (x^4-16)
3 log 2x^4(x-15) - 1/3log 12 (x^4-16)

Some parentheses would make this more readable, as would the inclusion of = for things that are equal.

Also, sqrt(x) = x1/2, not x1/3.

Mark44 said:
Some clarification, please. The numerator of what you wrote is
2x4 * (x - 15)3.

If that isn't what you meant, please use parentheses or brackets to make it clearer.

Some parentheses would make this more readable, as would the inclusion of = for things that are equal.

Also, sqrt(x) = x1/2, not x1/3.

1) yeah, that's what i meant to say, but i didn't know how to do the exponents

2) ok okay thanks! did I solve the problem right because I'm not very sure :\

Starting here --
log [ ( 2x^4 * (x-15)^3) / (12 sqrt (x^4-16) ]
= log [ 2x^4 * (x-15)^3] - log[12 * (x^4-16)^(1/2)]
Can you continue?

Notice that I added * to indicate multiplication. I think that's what you intended, but am not sure.

There are some properties of logs that you either don't know or aren't using, such as log(AB) = log(A) + log(B), assuming both A and B are positive.

Mark44 said:
Starting here --
log [ ( 2x^4 * (x-15)^3) / (12 sqrt (x^4-16) ]
= log [ 2x^4 * (x-15)^3] - log[12 * (x^4-16)^(1/2)]
Can you continue?

Notice that I added * to indicate multiplication. I think that's what you intended, but am not sure.

There are some properties of logs that you either don't know or aren't using, such as log(AB) = log(A) + log(B), assuming both A and B are positive.

after continuing would it be:

[log 2x^4 + log (x-15)^3] - [log 12 + log (x^4-16)^1/2]

and then use the power rule?

[4 log 2x + 3 log (x-15)] - [log 12 + 1/2 log (x^4-16)]

lolilovepie said:
after continuing would it be:

[log 2x^4 + log (x-15)^3] - [log 12 + log (x^4-16)^1/2]

and then use the power rule?

[4 log 2x + 3 log (x-15)] - [log 12 + 1/2 log (x^4-16)]

It's still not completely expanded.

## 1. How do you expand a log as a sum or difference of logs?

To expand a log as a sum or difference of logs, you can use the following rules:

• The sum of two logs is equal to the log of their product: log(ab) = log(a) + log(b)
• The difference of two logs is equal to the log of their quotient: log(a/b) = log(a) - log(b)

## 2. What is the purpose of expanding a log as a sum or difference of logs?

The purpose of expanding a log as a sum or difference of logs is to simplify the expression and make it easier to work with. This is especially useful when dealing with complex logarithmic equations.

## 3. Can you provide an example of expanding a log as a sum or difference of logs?

Sure, for example, log(2x) can be expanded as log(2) + log(x). Similarly, log(10/5) can be expanded as log(10) - log(5).

## 4. Are there any specific rules or restrictions when expanding a log as a sum or difference of logs?

Yes, when expanding a log as a sum or difference of logs, it is important to remember that only the bases of the logarithms should be the same. Also, the coefficients of the logs should not be included in the expansion.

## 5. Is it possible to expand a log with more than two terms as a sum or difference of logs?

No, a log can only be expanded as a sum or difference of logs when it has two terms. If there are more than two terms, the expression cannot be simplified further using these rules.