How do I solve for $t$ and $s$ in $y = x^{-t/s}$?

Question

I have $$y = x^{-t/s}$$

How do I solve for $t$ and $s$ in terms of the other variables?

It would be useful if you provided what you have tried. That way users can determine the level at which to answer your question. — Jeel Shah, Dec 12 '13 at 23:12

score 4 · Accepted Answer · answered Dec 12 '13 at 22:46

4

Take the logarithm of both sides of the equation.

$$y = x^{-t/s} \iff \ln y = \ln\Big(x^{-t/s}\Big)$$

Now use a key property of logarithms to extract $-t/s$ from the exponent of $x$: $$\ln(a^b) = b\ln a$$

answered Dec 12 '13 at 22:46

amWhy

So than -t/s = y/(ln(x)) ? – Squirrl Dec 12 '13 at 22:52
$-\dfrac ts = \dfrac{\ln y}{\ln x}$ – amWhy Dec 12 '13 at 22:53
Oh because I have to take the log from both sides?.. duh ;) – Squirrl Dec 12 '13 at 22:54
Exactly! That's right. – amWhy Dec 12 '13 at 22:54
THANK U. Is ln the same as log? – Squirrl Dec 12 '13 at 22:55
Yes, $\log$ is just fine, and any base will do, as long as it's the same base on each side of the equation. I just used the natural log (base $e$). – amWhy Dec 12 '13 at 22:56
Thanks again, I'll accept in a couple mins when they let me ^_^ – Squirrl Dec 12 '13 at 22:57
You're welcome. Glad to help! – amWhy Dec 12 '13 at 23:14
@amWhy: Nice feedback ... +1 – Amzoti Dec 13 '13 at 02:21

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