# How do you simplify log_5 (17/8) log_5 (51/16)?

Feb 4, 2017

${\log}_{5} \left(\frac{17}{8}\right) {\log}_{5} \left(\frac{51}{16}\right) = 1.18862$

#### Explanation:

${\log}_{5} \left(\frac{17}{8}\right) {\log}_{5} \left(\frac{51}{16}\right)$

= ${\log}_{5} \left(\frac{17}{8} \times \frac{51}{16}\right)$

= ${\log}_{5} \left(\frac{17 \times 17 \times 3}{8 \times 8 \times 2}\right)$

= $2 {\log}_{5} \left(\frac{17}{8}\right) + {\log}_{5} \left(\frac{3}{2}\right)$ as $\log {a}^{n} b = n \log a + \log b$

Now as ${\log}_{5} u = \log \frac{u}{\log} 5$, this can be written as

$\frac{2 \log \left(\frac{17}{8}\right) + \log \left(\frac{3}{2}\right)}{\log} 5$

= $\frac{2 \log \left(2.125\right) + \log \left(1.5\right)}{\log} 5$

= $\frac{2 \times 0.32736 + 0.17609}{0.69897}$

= $1.18862$