Question

Given that log5 3=0.683 and log5 7=1.209.without using calculator,evaluate

(a) log5 1.4

b)log7 75

c) log3 125

Answer 1

Did you notice that 1.4 = 7/5 ?

log₅ 1.4
= log₅ (7/5)
= log₅ 7 - log₅ 5
= 1.209 - 1
= .209

A property of logs that comes in handy here is
log_a b = 1/log_b a

b) log₇ 75
= log₇ (3*5^2)
=log₇ 3 + 2log₇ 5
= (log₅ 3) / (log₅ 7) + 2/log₅ 7)
= .683/1.209 + 2/1.209
= 2.683/1.209
= 2683/1209 ----> At this point I used Calculator to get appr 2.219


c) log₃ 125
= log₃ 5^3
= 3 log₃ 5
= 3 (1/log₅ 3)
= 3/.683