Some of the rules used in logarithms,
log m + log n = log (m x n)
log m - log n = log (m / n)
log m^{n }= n log m
log_{a} a = 1
Without using a calculator, simplify.
Problem 1 :
log 8 / log 2
Solution :
Simplifying the numerator and denominators separately.
log 8 = log2^{3}
= 3log2
= 3log2/log2
= 3(1)
= 3
Problem 2 :
log 9 / log 3
Solution :
Simplifying the numerator, we get
= log 9 / log 3
= log 3^{2}
= 2 log 3
Applying it in the given question, we get
= 2 log 3/log3
= 2(1)
= 2
Problem 3 :
log 4 / log 8
Solution :
= log 4 / log 8
= log 2^{2}/log 2^{3}
= 2/3
Problem 4 :
log 5 / log (1/5)
Solution :
= log 5 / log (1/5)
= log 5 / log 5^{-1}
= log 5 / -1log 5
= -(log 5/log 5)
= -1
Problem 5 :
log (0.5) / log 2
Solution :
= log (0.5) / log 2
= log (5/10)/ log 2
log (5/10) x 2
= log5/5
= 1
Problem 6 :
log 8 / log (0.25)
Solution :
= log 8 / log (0.25)
log 8 = log 2^{3}
= 3 log 2 ----(1)
log (0.25) = log (25/100)
= log 1/4
= log 1/2^{2}
= log 2^{-2}
= -2 log 2 ----(2)
(1) / (2)
= 3 log 2/(-2 log 2)
= -3/2
Problem 7 :
log 2^{b} / log 8
Solution :
log_{b} a = log a / log b
= log 2^{b} / log 8
= log_{8} 2^{b}
= b × log_{8} 2
= b × log_{2}^{3} 2
= b × 1/3 × log_{2}2
= b/3
Problem 8 :
log 4 / log 2^{a}
Solution :
= log 4 / log 2^{a}
log 4 = log 2^{2}
= 2 log 2 ---(1)
log 2^{a} = a log 2 ---(2)
(1) / (2)
= 2 log 2/a log 2
= 2/a
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM