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

Answer:

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

Step-by-step explanation:

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

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

$\displaystyle \ln A - \ln B = \ln\left(\frac{A}{B}\right)$.

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

First expression:

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

Second expression:

$\begin{aligned}&\ln 3+ \ln (1.4) - \ln 16 \\ &= \ln(3 \times 1.4) - \ln (16) \\ &= \ln\left(\frac{3 \times 1.4}{16}\right) = \ln(0.2625) \end{aligned}$.

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.