Without using log tables, prove that 2/5 < log10 3 < 1/2.

1.	Without using log tables, prove that 2/5 < log10 3 < 1/2.
Answer» We have to prove that, 2/5 < log₁₀ 3 < 1/2 i.e., to prove that 2/5 < log₁₀ 3 and log₁₀ 3< 1/2 i.e., to prove that 2 < 5 log₁₀ 3 and 2 log₁₀ 3 < 1 i.e., to prove that 2 log₁₀ 10 < 5 log₁₀3 and 2 log₁₀ 10 …… [∵ log_a a = 1] i.e., to prove that log₁₀ 10² < log₁₀ 3⁵ and log₁₀ 3² < log₁₀ 10 i.e., to prove that 10² < 3⁵ and 3² < 10 i.e., to prove that 100 < 243 and 9 < 10 which is true ∴ 2/5 < log₁₀3 < 1/2

Answer»

We have to prove that, 2/5 < log₁₀ 3 < 1/2

i.e., to prove that 2/5 < log₁₀ 3 and log₁₀ 3< 1/2

i.e., to prove that 2 < 5 log₁₀ 3 and 2 log₁₀ 3 < 1

i.e., to prove that 2 log₁₀ 10 < 5 log₁₀3 and 2 log₁₀ 10 ……

[∵ log_a a = 1]

i.e., to prove that log₁₀ 10² < log₁₀ 3⁵ and log₁₀ 3² < log₁₀ 10

i.e., to prove that 10² < 3⁵ and 3² < 10

i.e., to prove that 100 < 243 and 9 < 10 which is true

∴ 2/5 < log₁₀3 < 1/2

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