1. ## Lovely Logarithms

Hello,
I tried to solve this question, but I got the conclusion that I couldn't.
Therefore I would appreciate your help

The problem is:

Let log10P = x, log10Q = y and log10R = z.
Express log10 (P/QR^3)^2 in terms of x, y and z.

Thanks.

2. Use this:

$\lg (a\cdot b)=\lg a+\lg b$
$\lg\left(\frac{a}{b}\right)=\lg a-\lg b$
$\lg a^n=n\lg a$

but how?

"How?" . . . Have you never worked a log problem? . . . ever?

Okay, just this once . . . I'll baby-step through it for you.

Since all the logs are base-ten, I'll drop the base . . .

Let: . $\log(P) = x,\;\;\log(Q)= y,\;\;\log(R)= z$

Express $\log\bigg(\frac{P}{QR^3}\bigg)^2$ in terms of $x, y\text{ and }z.$

We have: . $\log\bigg(\frac{P}{QR^3}\bigg)^2 \;=\;2\bigg[\log\left(\frac{P}{QR^3}\right)\bigg]$

. . $=\;2\bigg[\log(P) - \log(QR^3)\bigg]$

. . $= \;2\bigg[\log(P) - \left\{\log(Q) + \log(R^3) \right\} \bigg]$

. . $=\;2\bigg[\log(P) - \log(Q) - \log(R^3)\bigg]$

. . $= \;2\bigg[\underbrace{\log(P)}_x - \underbrace{\log(Q)}_y - 3\underbrace{\log(R)}_z\bigg]$

. . $= \;2\left(x - y - 3z\right) \quad\hdots\quad\text{or: }\;2x - 2y - 6z$

5. Thanks