Question

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b' : log 5.4 

Answer 1

Given that log₁₀2 = a and log₁₀3 = b
log 5.4 
= log`54/10`
= log`( 2 xx 3 xx 3 xx 3)/10`
= log( 2 x 3 x 3 x 3 ) - log₁₀10   ...[ log_am = log_an = log_a`(m/n)` ]
= log₁₀2 + log₁₀3³ - log₁₀10   ...[log_amn = log_am + log_an]
= log₁₀2 + 3log₁₀3 - log₁₀10  ...[ nlog_am = log_amⁿ ]
= log₁₀2 + 3log₁₀3 - 1           ...[ ∵ log₁₀10 = 1]
= a + 3b - 1                            ...[ ∵ log₁₀2 = a and log₁₀3 = b ]