Manipulation of Logarithms

Level pending

$\begin{cases} \log(AB) = X \\ \log(A)\log(B) = Y \\ \log_{A}(B) - \log_{B}(A) = Z\end{cases}$

Given the above, find the value of: $\log(B) - \log(A)$ in terms of $X$ and $Y$ and $Z$ .

\frac{XY}{Z}

\frac{XZ}{Y}

\frac{X}{YZ}

\frac{ZY}{X}

1 solution

Kevin Lee
Jul 24, 2016

Given: $log_{A}(B) - log_{B}(A) = Z$ ... this can be manipulated into:

$\frac{log(B)}{log(A)} - \frac{log(A)}{log(B)}$ ... get a common denominator ...
$\frac{(log(B))^2}{log(A)log(B)} - \frac{(log(A))^2}{log(B)log(A)}$ ...
Let $x = log(B)$ and let $y = log(A)$ and substitute:
$\frac{x^2-y^2}{xy}$ ... factor:
$\frac{(x+y)(x-y)}{(xy)}$ ... replace known values:
$\frac{X(x-y)}{Y} = Z$ ... solve for $(x-y)$ ... (keep in mind: $log(AB) = log(A) + log(B) = X$ )
$(x-y) = \frac{YZ}{X}$