Question

Without using log table, prove the following:
i) $\frac{3}{10} \ \textless \ \log_{10} 2 \ \textless \ \frac{1}{3}$
ii) $\frac{2}{3} \ \textless \ \log_{10} 5 \ \textless \ \frac{3}{4}$

Answer 1

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Answer 2

Answer:

3/10 < log2 < 1/3

2/3 < log 5 < 3/4

Step-by-step explanation:

To be proved

3/10 < log2 < 1/3

take one by one

3/10 < log2

if 3 < 10 log 2

if 3 < log 2¹⁰

if log 10³ < log 1024

if log 1000 < log (1024)

which is true

hence

3/10 < log2

log2 < 1/3

if 3log2 < 1

if log 2³ < log 10

if log 8 < log 10

which is true

Hence

3/10 < log2 < 1/3

2/3 < log 5 < 3/4

2/3 < log 5

if 2 < 3 log 5

if log 10² < log 5³

if log 100 < log 125

which is true

log 5 < 3/4

if  4log5 < 3

if log 5⁴ < log 10³

if log 625 < log 1000

which is true

hence

2/3 < log 5 < 3/4