# Math Help - logs again

1. ## logs again

simplify log(base4/9)(27/16) without a calculator

i've tried everything i could think of without a calculator, so here i am.

simplify log(base4/9)(27/16) without a calculator

i've tried everything i could think of without a calculator, so here i am.
$\frac{27}{16} = \frac{3}{4}\cdot\frac{9}{4}$

$= \frac{3}{4}\left(\frac{4}{9}\right)^{-1}$.

So $\log_{\frac{4}{9}}\frac{27}{16} = \log_{\frac{4}{9}}\left[\frac{3}{4}\left(\frac{4}{9}\right)^{-1}\right]$

$= \log_{\frac{4}{9}}\frac{3}{4} + \log_{\frac{4}{9}}\left(\frac{4}{9}\right)^{-1}$

$= \log_{\frac{4}{9}}\frac{3}{4} - 1$

$= \log_{\frac{4}{9}}{3} - \log_{\frac{4}{9}}{4} - 1$.

3. Originally Posted by Prove It
$\frac{27}{16} = \frac{3}{4}\cdot\frac{9}{4}$

$= \frac{3}{4}\left(\frac{4}{9}\right)^{-1}$.

So $\log_{\frac{4}{9}}\frac{27}{16} = \log_{\frac{4}{9}}\left[\frac{3}{4}\left(\frac{4}{9}\right)^{-1}\right]$

$= \log_{\frac{4}{9}}\frac{3}{4} + \log_{\frac{4}{9}}\left(\frac{4}{9}\right)^{-1}$

$= \log_{\frac{4}{9}}\frac{3}{4} - 1$

$= \log_{\frac{4}{9}}{3} - \log_{\frac{4}{9}}{4} - 1$.
silly me, i was trapped in the thought of solving it! thanks for the enlightenment on my silly misunderstanding

simplify log(base4/9)(27/16) without a calculator

i've tried everything i could think of without a calculator, so here i am.
Here's an alternative approach.

Let $x = \log_{\frac49}\left(\frac{27}{16}\right)$

Then $\left(\frac49\right)^x = \frac{27}{16}$

$\Rightarrow \left(\left(\frac23\right)^2\right)^x = \frac{3^3}{2^4}$

$\Rightarrow \left(\frac23\right)^{2x}=\frac12\left(\frac23\rig ht)^{-3}$

$\Rightarrow 2x = \log_{\frac23}\left(\frac12\right)-3\log_{\frac23}\left(\frac23\right)$
$=-\log_{\frac23}(2)-3$
$\Rightarrow x = -\tfrac12(\log_{\frac23}(2)+3)$

...but whether you think this answer is any simpler is a matter for debate!

If you want to get rid of fractional bases, you can express this in terms of $\log_2$ using the formula
$\log_{\frac23}(2) = \frac{\log_22}{\log_2(\tfrac23)}$
$=\frac{1}{1-\log_23}$
which then gives:
$x = \frac{4-3\log_23}{2(\log_23-1)}$
but I don't think you can rid of logs altogether!