Some clarification, please. The numerator of what you wrote islolilovepie said:Homework Statement
Completely expand as a sum/difference of logs
log [ ( 2x^4(x-15)^3) / (12 sqrt (x^4-16) ]
lolilovepie said:Homework Equations
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)
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.
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.
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)]
To expand a log as a sum or difference of logs, you can use the following rules:
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.
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).
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.
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.