​ Without using log table, prove the following: i. 1/4 < log10 2 < ​1/3 ​ ii. 2/5 < log10 3 < 1/2

Question

Without using log table, prove the following: 

i. \(\cfrac{1}{4}\) < log₁₀ 2 < \(\cfrac{1}{3}\)

ii. \(\cfrac{2}{5}\) < log₁₀ 3 < \(\cfrac{1}{2}\)

iii. \(\cfrac{3}{10}\) < log₁₀ 2 < \(\cfrac{1}{3}\)

iv. \(\cfrac{2}{3}\)< log₁₀ 5 < \(\cfrac{3}{4}\)

Answer 1

i. We have to prove that,  \(\cfrac{1}{4}\) < log₁₀ 2 < \(\cfrac{1}{3}\)

i.e., to prove that 10 < 2⁴ and 2³ < 10

i.e., to prove that 10 < 16 and 8 < 10

which is true