Question

Log root 27 + log8 + log root 1000 /log120=3\2

Answer 1

Consider the LHS as:

$\frac{\log {\sqrt27} + \log 8 + \log \sqrt{1000}}{\log{120}}$

= $\frac{\log {27^\frac{1}{2}} + \log 2^3 + \log {1000^\frac{1}{2}}}{\log{(2 \times 2 \times 2 \times 3 \times 5)}}$

Using the laws of logarithmic which states, \log m^n = n \log m

$= \frac{\frac{1}{2} \log(3^3) + \log(2^3) + \frac{1}{2} \log(10^3)}{\log(2^3 \times 3 \times 5)}$

=$\frac{\frac{3}{2} \log3 + 3 \log(2) + \frac{3}{2} \log(10)}{\log(2^3)+\log3+ \log 5}$

= $\frac{\frac{3}{2} \log3 + 3 \log(2) + \frac{3}{2} \log(2 \times 5)}{3 \log2+\log3+ \log 5}$

$= \frac{\frac{3}{2} \log3 + 3 \log2 + \frac{3}{2} (\log2 + \log 5)}{3 \log2+\log3+ \log 5}$

=$\frac{\frac{3}{2} \log3 + 3 \log2 + \frac{3}{2} \log2 +\frac{3}{2} \log 5}{3 \log2+\log3+ \log 5}$

=$\frac{\frac{3}{2} \log3 + \frac{9}{2} \log2 +\frac{3}{2} \log 5}{3 \log2+\log3+ \log 5}$

=$\frac{3}{2}\frac{( \log3 + 3 \log2 + \log 5)}{3 \log2+\log3+ \log 5}$

= $\frac{3}{2}$

=RHS

Hence, proved.

Answer 2

hope it helps to u to