Question

3.

Which of the following numbers is/are positive ?
1
(A) In 2 + In 3 + In 0.161
(B) In 3 + In(1.4) - In 16​

Answer 1

Neither expression (A) nor expression (B) would be positive. A calculator is not required.

Explanation:

Assume that
`A` and
`B` are both positive real numbers. Consider the property of logarithms with the same base:

`\ln A + \ln B = \ln(A \cdot B)`.

`\displaystyle \ln A - \ln B = \ln\left((A)/(B)\right)`.

Apply these two properties to combine the logarithms in each expression into a one.

First expression:

`\begin{aligned}&\ln 2+ \ln 3 + \ln 0.161 \\ &= \ln(2 * 3) + \ln (0.161) \\ &= \ln(2 * 3 * 0.161) = \ln(0.966) \end{aligned}`.

Second expression:

`\begin{aligned}&\ln 3+ \ln (1.4) - \ln 16 \\ &= \ln(3 * 1.4) - \ln (16) \\ &= \ln\left((3 * 1.4)/(16)\right) = \ln(0.2625) \end{aligned}`.

The logarithm of a real number is positive if and only if that number is greater than one.

Neither
`0.966` nor
`0.2625` is greater than one. Therefore, neither of their respective logarithms would be positive.