Question

If $x$ is a real number such that $\log_3 5 = \log_5(2 + x)$, then which of the following is true?

Accepted Answer

D. We know $1 < \log_3 5 < 2$, because $3^1 = 3 < 5 < 9 = 3^2$. Let $k = \log_3 5$, so $1 < k < 2$. Given $\log_5(2 + x) = k \implies 2 + x = 5^k$. Since $1 < k < 2$: $$5^1 < 5^k < 5^2 \implies 5 < 2 + x < 25$$ $$3 < x < 23$$ Thus, $3 < x < 23$.

Logarithmic Range for x

Explanation

Practise this under exam conditions