Question

Which of the following statements is/are true?

1. If $lo{g}_{m}a<b\Rightarrow a>m^b;whenm>1$

2. If $lo{g}_{m}a<b\Rightarrow a>m^{b}when0<m<1$

3. If $lo{g}_{m}a>b\Rightarrow a<m^{b}when0<m<1$

4. If $lo{g}_{m}a>b\Rightarrow a<m^{b}whenm>1$

• A. Only 1 and 3
• B. Only 2 and 3
• C. Only 2 and 4
• D. Only 1 and 4

Answer 1

The correct option is B

Only 2 and 3

Statement 1: $Iflo{g}_{m}a<bwhenm>1$

In the previous question we have seen when $m>1,lo{g}_{m}y$ is an increasing function.

So, when value of x increases logarithm function values increases.

When $m>1$

$iflo{g}_m{x}_1<lo{g}_m{x}_2$

$x_1<x_2$

Right hand side of inequality can be written as

$lo{g}_{m}a<lo{g}_m{m}^b$

So, $a<m^b\{form>1\}$

So,statement 1 is wrong.

Statement 2 : if $lo{g}_{m}a<b$ when $0<m<1$

When $0<m<1,lo{g}_{m}x$ is a decreasing function when value of x increases,

logarithm function value decreases.

RHS of inequality can be written as

$lo{g}_{m}a<lo{g}_m{m}^b$

When $0<m<1$

$a>m^b$

So statement 2 is correct

Statement 3: $lo{g}_{m}a>b$ when $0<m<1$

We have already seen when $0<m<1,lo{g}_{m}x$ is a decreasing function.

If $lo{g}_{m}a>lo{g}_m{m}^b$

$a<m^b$

So,statement 3 is correct

Statement 4: $lo{g}_{m}a>bwhenm>1$

We have seen when $m>1.lo{g}_{m}x$ is an increasing function. If value of x increases

$lo{g}_{m}x$ value also increases.

If $lo{g}_{m}a>lo{g}_m{m}^b$

$a>m^b$

so, statement 4 is incorrect.

Only statement 2 and statement 3 are correct.