Question

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b': log ${\displaystyle 2\frac{1}{4}}$

Answer 1

Given that log₁₀2 = a and log₁₀3 = b
log ${\displaystyle 2\frac{1}{4}}$
= log ${\displaystyle \left(\frac{9}{4}\right)}$
= log ${\displaystyle {\left(\frac{3}{2}\right)}^2}$         
= 2log${\displaystyle \left(\frac{3}{2}\right)}$                ...[ nlog_am = log_amⁿ ]
= 2( log3 - log2 )         ...[ log_am - log_an = log_a ${\displaystyle \frac{m}{n}}$ ]
= 2( b - a )                   ...[ ∵ log₁₀2 = a and log₁₀3 = b ]
= 2b - 2a