Question

Without using log table, prove the following:
i) $\frac{1}{4} \ \textless \ \log_{10} 2 \ \textless \ \frac{1}{3}$
ii) $\frac{2}{3} \ \textless \ \log_{10} 3 \ \textless \ \frac{1}{2}$

Answer 1

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Answer 2

Answer:

1/4 < log2 < 1/3

1/3 < log 3 < 1/2

Step-by-step explanation:

To be proved

1/4 < log2 < 1/3

take one by one

1/4 < log2

if 1 < 4 log 2

if log 10 < log 2⁴

if log 10 < log 16

which is true

hence

1/4 < log2

log2 < 1/3

if 3log2 < 1

if log 2³ < log 10

if log 8 < log 10

which is true

Hence

1/4 < log2 < 1/3

correct Question is :

1/3 < log 3 < 1/2

1/3 < log 3

if 1 < 3 log 3

if log 10 < log 3³

if log 10 < log 27

which is true

log 3 < 1/2

if  2log3 < 1

if log 3² < log 10

if log 9 < log 10

which is true

hence

1/3 < log 3 < 1/2