This is the exercise, there are no clues in the book about it.
$$
40\log_{4x}x^\frac{1}{2}-14\log_{16x}x^3=-\log_{\frac{1}{2}x}x^2
$$
Solutions given by the book: $x=1; x=4; x=\frac{\sqrt{2}}{2}.$
And this is what I did so far:
- conditions for existence:
$$
\Biggl\{
\begin{eqnarray}
x &\gt& 0.\\
x &\ne& \frac 14; x \ne \frac{1}{16}; x \ne 2.
\end{eqnarray}
$$ - I simplified some exponent in the arguments:
$$
20\log_{4x}x-42\log_{16x}x+2\log_{\frac{1}{2}x}x=0
$$ - Change of bases:
$$
\frac{20}{\log_x4x}-\frac{42}{\log_x16x}+\frac{2}{\log_x\frac{1}{2}x}=0
$$ - Observing that: $\log_xnx=1+\log_xn$
- least common multiple:
$$
\frac{20(1+\log_x16)(1+\log_x\frac12)-42(1+\log_x4)(1+\log_x\frac12)+2(1+\log_x4)(1+\log_x16)}{(1+\log_x16)(1+\log_x4)(1+\log_x\frac12)}=0
$$ - denominator can be toggled, as for conditions for existence
- then I lost confidence in what I was doing…
Any clue is welcome, thanks :]
Best Answer
We can use that
$$\log_a^b=\frac{\log a}{\log b}$$
therefore
$$40\log_{4x}x^\frac{1}{2}-14\log_{16x}x^3=-\log_{\frac{1}{2}x}x^2$$
$$40\frac{\log x^\frac{1}{2}}{\log {4x}}-14\frac{\log x^3}{\log {16x}}=-\frac{\log x^2}{\log {\frac{1}{2}x}}$$
$$20\frac{\log x}{\log {x}+\log 4}-42\frac{\log x}{\log {x}+\log 16}=-2\frac{\log x}{\log {x}+\log \frac12}$$
and eliminating $\log x \neq 0$ (which is a solution)
$$\frac{10}{\log {x}+2\log 2}-\frac{21}{\log {x}+4\log 2}=-\frac{1}{\log {x}-\log 2}$$
then let $y=\log x$ and $a=\log 2$ to obtain
$$\frac{10}{y+2a}-\frac{21}{y+4a}+\frac{1}{y-a}=0$$
$$\frac{10(y+4a)(y-a)-21(y+2a)(y-a)+(y+2a)(y+4a)}{(y+2a)(y+4a)(y-a)}=0$$
$$\frac{10(y^2+3ay-4a^2)-21(y^2+ay-2a^2)+(y^2+6ay+8a^2)}{(y+2a)(y+4a)(y-a)}=0$$
$$\frac{-10 y^2+15 ay+10a^2}{(y+2a)(y+4a)(y-a)}=0$$
$$\frac{-5(y-2a)(2y+a)}{(y+2a)(y+4a)(y-a)}=0$$
that is
$\log x = 2\log 2$
$2\log x = -\log 2$