For positive numbers b, r, and s, where $b \neq 1$ , $\log _ { b } ( r + s ) = \log _ { b } r + \log _ { 3 } s$ .

Question

Quizplus · Accepted Answer

The given equation can be written as:
$$
\log_b(r+s) = \frac{\log_r(r+s)}{\log_r b} = \frac{\log_r r}{\log_r b} + \frac{\log_3 s}{\log_r b} = 1 + \frac{\log_3 s}{\log_r b}
$$
On the other hand,
$$
\log_b(r+s) = \frac{\log_b(r+s)}{1} = \frac{\log_b r + \log_3 s}{1} = \frac{\log_b r}{1} + \frac{\log_3 s}{1} = \log_b r + \log_3 s
$$
Hence, the given equation is true if and only if $\frac{\log_3 s}{\log_r b}=0$, which is not necessarily true for all choices of $r$ and $s$. Therefore, the answer is B.

For Positive Numbers B, R, and S, Where b≠1b \neq 1b=1

For Positive Numbers B, R, and S, Where $b \neq 1$