Logarithm Multiple Choice Question

[veronicadeno]

I'm studying for an advanced math 12 regional exam and got the following multiple choice question wrong.

Which expression is equivalent to 2log(4/x), x(not equal to)0, for all possible values of x?

The correct answer is log16-2logx. I chose (log4-logx)² by doing this;
2log(4/x)
log(4/x)²
(log4-logx)²

Can anyone show me the correct way to solve this problem?

 

[lookagain]

I'm studying for an advanced math 12 regional exam and got the following multiple choice question wrong.

Which expression is equivalent to 2log(4/x),
x(not equal to)0, for all possible values of x?
Not only that, but x shouldn't be negative, either.

The correct answer is log16-2logx. I chose (log4-logx)² by doing this;
2log(4/x)
log(4/x)^{2
No, those two expressions are not equivalent to each other if x < 0.}



(log4-logx)²


And, anyway, log[(4/x)^2] is not equal to [log(4/x)]^2,

of which the latter is equal to [log(4) - log(x)]^2.

\(\displaystyle 2log \bigg(\dfrac{4}{x} \bigg) = \)

\(\displaystyle 2[log(4) - log(x)] = \)

\(\displaystyle 2log(4) - 2log(x) = \)

\(\displaystyle log(4^2) - 2log(x) = \)

\(\displaystyle log(16) - 2log(x)\)