Question

Answer the following:

Without using log tables, prove that `2/5 < log_10 3 < 1/2`

Answer 1

We have to show that, `2/5 < log_10 3 < 1/2`

i.e., to show that,

`2/5 < log_10 3` and `log_10 3 < 1/2`

i.e., to show that,

2 < 5log₁₀3 and 2 log₁₀3 < 1

i.e., to show that,

2 log₁₀10 < 5 log₁₀3 and 2 log₁₀3 < log₁₀10  ...[∵ log₁₀10 = 1]

i.e., to show that,

log₁₀10² < log₁₀3⁵ and log₁₀3² < log₁₀10

i.e., to show that,

10² < 3⁵ and 3² < 10

i.e., to show that,

100 < 243 and 9 < 10

which is true

∴ `2/5 < log_10 3 < 1/2`.