# How do you solve log_5x + log_3 x=1?

May 7, 2018

$x = 1.9211$

#### Explanation:

${\log}_{5} x + {\log}_{3} x = 1$ can be written as

$\log \frac{x}{\log} 5 + \log \frac{x}{\log} 3 = 1$

or $\log 3 \log x + \log 5 \log x = \log 3 \log 5$

or $\log x = \frac{\log 3 \log 5}{\log 3 + \log 5}$

or $x = {10}^{\frac{\log 3 \log 5}{\log 15}}$

= ${10}^{\frac{0.4771 \cdot 0.6990}{1.1761}}$

= ${10}^{0.2836}$

= $1.9211$