How do you simplify ${log}_{5} (\frac{17}{8}) {log}_{5} (\frac{51}{16})$ $log_5 (17/8) log_5 (51/16)$ ?

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Answer 1 · 2017-02-04T07:15:59

${log}_{5} (\frac{17}{8}) {log}_{5} (\frac{51}{16}) = 1.18862$ $log_5(17/8)log_5(51/16)=1.18862$

${log}_{5} (\frac{17}{8}) {log}_{5} (\frac{51}{16})$ $log_5(17/8)log_5(51/16)$

= ${log}_{5} (\frac{17}{8} \times \frac{51}{16})$ $log_5(17/8xx51/16)$

= ${log}_{5} (\frac{17 \times 17 \times 3}{8 \times 8 \times 2})$ $log_5((17xx17xx3)/(8xx8xx2))$

= $2 {log}_{5} (\frac{17}{8}) + {log}_{5} (\frac{3}{2})$ $2log_5(17/8)+log_5(3/2)$ as ${log a}^{n} b = n log a + log b$ $loga^nb=nloga+logb$

Now as ${log}_{5} u = \frac{log u}{log 5}$ $log_5u=logu/log5$ , this can be written as

$\frac{2 log (\frac{17}{8}) + log (\frac{3}{2})}{log 5}$ $(2log(17/8)+log(3/2))/log5$

= $\frac{2 log (2.125) + log (1.5)}{log 5}$ $(2log(2.125)+log(1.5))/log5$

= $\frac{2 \times 0.32736 + 0.17609}{0.69897}$ $(2xx0.32736+0.17609)/(0.69897)$

= $1.18862$ $1.18862$

How do you simplify log5(178)log5(5116)log_5 (17/8) log_5 (51/16)?