First slide

Types of real valued functions XI

Question

If $2^{x + y} = 6^{y} and 3^{x - 1} = 2^{y + 1}$ then the value of $(\log 3 - \log 2) / (x - y)$ is

Moderate

A

1
B

$\log_{2} 3 - \log_{3} 2$
C

$\log (3 / 2)$
D

none of these

Solution

Taking log, we have
$\begin{array}{ll} (x + y) \log 2 = y (\log 2 + \log 3) \\ ∴ xlog 2 = ylog 3 \\ or \frac{x}{\log 3} = \frac{y}{\log 2} = \frac{x - y}{\log 3 - \log 2} = λ (say) -- (i) \end{array}$
$\begin{array}{l} also (x - 1) \log 3 = (y + 1) \log 2 \\ now xlog 3 - ylog 2 = \log 3 + \log 2 \end{array}$
Using Eq. (i), we get
$\begin{array}{l} λ [(\log 3)^{2} - (\log 2)^{2}] = \log 3 + \log 2 \\ λ = \frac{1}{\log 3 - \log 2} \\ ∴ \frac{1}{λ} = \frac{1}{\log 3 - \log 2} \Rightarrow λ = \log \frac{3}{2} \end{array}$

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