Logarithm MCQs

$text{log } Big(frac{a + b}{3}Big) = frac{1}{2} (text{log } a + text{log } b)$

$therefore text{log } Big(frac{a + b}{3}Big) = text{ log } (ab)^frac{1}{2} text{ or } frac{a + b}{3} = (ab)^frac{1}{2}$

$text{ or } (a +b )^2 = 9ab Rightarrow a^2 + b^2 = 7ab$

$begin{aligned} & text{ log } 11 - text{ log } 5 + text{ log } 2 + text{ log } 7 - text{ log } 3 \ &- text{ log } 2 - text{ log } 11 + text{ log } 5 + text{ log }3 end{aligned}$

$therefore text{ log } 7$

$text{ log}_b a times text{ log}_y x = text{ log }_c a$

$text{ log}_b a times text{ log}_c b = text{ log }_c a$

$therefore x = b, y = c$

$begin{aligned} &log frac{1}{70} = log 1 - log 70 &quadquad = - log 70 &quadquad = - log (7 times 10) &quadquad = - (log 7 + log 10) &quadquad = - log 7 - log 10 &quadquad = - a - 1 = - (1 + a) &quadquad [because log 7 = a] end{aligned}$

$begin{aligned} log x &= bar 3.0385 = - 3.0385 quadquad; &= - 6 + 3.0385 quadquad; &= - 6 log 10 + 3.0385 quadquad; &=log 10^{-6} + log (1092) quadquad; &=log (1092 times 10^{-6}) quadquad; &= log (0.001092) Rightarrow x &= 0.001092 end{aligned}$

$begin{aligned} &frac{1}{3} log_{10} 125 - 2 log_{10} 4 + log_{10} 32 \ &= log_{10} (125)^{1/3} - log_{10} 4 ^2 + log_{10}^{32} \ &Big[because log m^n = n log mBig] \ &=log_{10} 5 - log_{10} 16 + log_{10} 32 \ &=log_{10} frac{5 times 32}{16} \ &=log_{10} 10 \ &= 1 \ &Big[ because log m times n = log m + log nBig] \ &Big[log frac{m}{n} = log m - log n Big] &end{aligned}$

Given , 0 < a < 1

Let $a = frac{1}{2} = 0.5$

$a^2 = frac{1}{4} = 0.25$

$frac{1}{a} = 2$

$log a = log frac{1}{2} = log_1 - log_2 = 0 - 0.30 = text{ negative}$

$therefore$ In ascending order of their values are

$log a, a^2, a, frac{1}{a}$

$begin{aligned} log_{10}125 &= log_{10}(5)^3 &= 3 log_{10}5 [because log m^n = n log m] &= 3 log_{10} Big(frac{10}{2}Big) &= 3[log_{10}10 - log_{10}2] &= 3 [1 - 0.30] &= 3 times 0.70 &= 2.10 end{aligned}$