Question

log1/2-log5/6+log3/4-log7/8( convert it into single logarithm)​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1.

$log(xy) = log(x) + log(y)$

2.

$\displaystyle \: log \bigg( \frac{x}{y} \bigg) = log(x) - log(y)$

TO SIMPLIFY

$\displaystyle \: log \bigg( \frac{1}{2} \bigg) - log \bigg( \frac{5}{6} \bigg) + log \bigg( \frac{3}{4} \bigg) - log \bigg( \frac{7}{8} \bigg)$

CALCULATION

$\displaystyle \: log \bigg( \frac{1}{2} \bigg) - log \bigg( \frac{5}{6} \bigg) + log \bigg( \frac{3}{4} \bigg) - log \bigg( \frac{7}{8} \bigg)$

$= \displaystyle \: \bigg[ \: log \bigg( \frac{1}{2} \bigg) + log \bigg( \frac{3}{4} \bigg) \bigg ] - \bigg[ log \bigg( \frac{5}{6} \bigg) + log \bigg( \frac{7}{8} \bigg) \bigg]$

$= \displaystyle \: log \bigg( \frac{1}{2} \times \frac{3}{4} \bigg) - log \bigg( \frac{5}{6} \times \frac{7}{8} \bigg)$

$= \displaystyle \: log \bigg( \frac{3}{8} \bigg) - log \bigg( \frac{35}{48} \bigg)$

$= \displaystyle \: log \bigg( \frac{3}{8} \times \frac{48}{35} \bigg)$

$= \displaystyle \: log \bigg( \frac{18}{35} \bigg)$