# Question #db07d

Feb 10, 2018

I got $\log \frac{21}{\log} \left(\frac{9}{7}\right) \approx 12.11439573$

#### Explanation:

Take the log of both sides

$\log \left({3}^{2 x - 1}\right) = \log \left({7}^{x + 1}\right)$

Use the exponent rule to bring down the exponents

$\left(2 x - 1\right) \left(\log 3\right) = \left(x + 1\right) \left(\log 7\right)$

Distribute

$2 x \log 3 - \log 3 = x \log 7 + \log 7$

Move the "x" terms to one side of the equation, constant terms to the other:

$2 x \log 3 - x \log 7 = \log 7 + \log 3$

Factor out the $x$ on the left side

$x \left(2 \log 3 - \log 7\right) = \log 7 + \log 3$

Isolate $x$ by dividing

$x = \frac{\log 7 + \log 3}{2 \log 3 - \log 7}$

If all you need is the decimal answer, put that into your calculator and you should get $x \approx 12.11439573$. If you need the exact answer, simplify the logs using laws of logarithims.

$x = \log \frac{7 \cdot 3}{\log} \left({3}^{2} / 7\right) = \log \frac{21}{\log} \left(\frac{9}{7}\right)$

(I don't think that simplifies any more...)