Logarithmic Equation with Two Variables

Question

If $\log_4 m + \log_4 n = \log_2(m + n)$ where $m$ and $n$ are positive real numbers, then which of the following must be true?

Accepted Answer

E. We have $\log_4(mn) = \log_2(m+n) \implies \frac{1}{2}\log_2(mn) = \log_2(m+n) \implies \sqrt{mn} = m+n$. Squaring both sides gives $mn = m^2 + 2mn + n^2 \implies m^2 + mn + n^2 = 0$. Since $m$ and $n$ are positive real numbers, $m^2 + mn + n^2 > 0$ always. Thus, no positive real values of $m$ and $n$ can satisfy the equation.

Answer

$\frac{1}{m} + \frac{1}{n} = 1$

Answer

$m = n$

Answer

$m^2 + n^2 = 1$

Answer

$\frac{1}{m} + \frac{1}{n} = 2$

Answer

No values of $m$ and $n$ can satisfy the given equation

Logarithmic Equation with Two Variables

Explanation

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