Question

log2(27)×log3(625)×log5(1/64)=_____. a. 72 b. -72 c. 24 d. -24

Answer 1

Given:

Log2(27)×log3(625)×log5(1/64)=_____.

To find:

Log2(27)×log3(625)×log5(1/64)=_____. a. 72 b. -72 c. 24 d. -24

Solution:

From given, we have,

Log2(27)×log3(625)×log5(1/64)=_____

$\log _2\left(27\right)\log _3\left(625\right)\log _5\left(\frac{1}{64}\right)\\\\=3\log _2\left(3\right)\log _3\left(625\right)\log _5\left(\frac{1}{64}\right)\\\\=3\cdot \:4\log _2\left(3\right)\log _3\left(5\right)\log _5\left(\frac{1}{64}\right)\\\\=3\cdot \:4\log _2\left(3\right)\log _3\left(5\right)\left(-6\log _5\left(2\right)\right)\\\\=-3\log _2\left(3\right)\cdot \:4\log _3\left(5\right)\cdot \:6\log _5\left(2\right)$

$=-72\log _2\left(3\right)\log _3\left(5\right)\log _5\left(2\right)\\\\=-72\cdot \frac{\ln \left(3\right)}{\ln \left(2\right)}\cdot \frac{\ln \left(5\right)}{\ln \left(3\right)}\log _5\left(2\right)\\\\=-72\cdot \frac{\ln \left(5\right)}{\ln \left(2\right)}\log _5\left(2\right)\\\\=-\frac{\ln \left(5\right)\cdot \:72\log _5\left(2\right)}{\ln \left(2\right)}$

$=-\frac{72\cdot \frac{\ln \left(2\right)}{\ln \left(e\right)}}{\ln \left(2\right)}\\\\=-\frac{72\ln \left(2\right)}{\ln \left(2\right)}\\\\=-72$

Option (b) -72 is correct