# How do you solve 7 * 3^t = 5 * 2^t?

May 3, 2018

By using Log function

#### Explanation:

$7 \cdot {3}^{t} = 5 \cdot {2}^{t}$
$\log \left(7 \cdot {3}^{t}\right) = \log \left(5 \cdot {2}^{t}\right)$
$\log \left(7\right) + \log \left({3}^{t}\right) = \log \left(5\right) + \log \left({2}^{t}\right)$
$\log \left(7\right) + t \log \left(3\right) = \log \left(5\right) + t \log \left(2\right)$
$\log \left(7\right) - \log \left(5\right) = t \log \left(2\right) - t \log \left(3\right)$
$\log \left(\frac{7}{5}\right) = t \left(\log \left(2\right) - \log \left(3\right)\right)$
$\log \left(\frac{7}{5}\right) = t \left(\log \left(\frac{2}{3}\right)\right)$
$\log \frac{\frac{7}{5}}{\log \left(\frac{2}{3}\right)} = t$
then make good use of your cal.