Question

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b': log 2.25

Answer 1

log₁₀2 = a and log₁₀3 = b
log 2.25

= log${\displaystyle \frac{225}{100}}$

= log${\displaystyle \frac{25\times 9}{25\times 4}}$

= log${\displaystyle \frac{25\times 9}{25\times 4}}$

= log${\displaystyle \left(\frac{9}{4}\right)}$

= ${\displaystyle {\log \left(\frac{3}{2}\right)}^2}$

= ${\displaystyle \log \left(\frac{3}{2}\right)}$                 ...[ nlog_am = log_amⁿ ]

= 2( log3 - log2 )        ...[ log_am - log_an = log_a${\displaystyle \left(\frac{m}{n}\right)}$]
= 2( b - a )                  ...[ ∵ log₁₀2 = a and log₁₀3 = b ]
= 2b - 2a