# How do you solve log_5 x + log_10 x = 4?

Mar 25, 2016

$x = 44.228$

#### Explanation:

As ${\log}_{b} x = {\log}_{a} \frac{x}{\log} _ a b$, we have

${\log}_{5} x + {\log}_{10} x = 4$ can be written as

${\log}_{10} \frac{x}{\log} _ 10 5 + {\log}_{10} x = 4$ or

${\log}_{10} \frac{x}{0.699} + {\log}_{10} x = 4$ or

${\log}_{10} x + 0.699 \times {\log}_{10} x = 4 \times 0.699$ or

${\log}_{10} x \left(1 + 0.699\right) = 4 \times 0.699$ or

${\log}_{10} x = \frac{4 \times 0.699}{1.699} = 1.6457$

$x = {10}^{1.6457} = 44.228$