The discussion focuses on the complete expansion of the logarithmic expression log [ ( 2x^4(x-15)^3) / (12 sqrt (x^4-16) ]. Participants clarify the application of logarithmic properties, specifically the quotient rule and power rule. The correct expansion involves separating the numerator and denominator using log properties, resulting in the expression [4 log 2 + 3 log (x-15)] - [log 12 + 1/2 log (x^4-16)]. The importance of proper notation and clarity in mathematical expressions is emphasized throughout the conversation.
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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)]