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Last updated date: 09th Dec 2023
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If ${x^{2a - 3}}{y^{2a}} = {x^{6 - a}}{y^{5a}}$ then the value of a $\log (\dfrac{x}{y})$ is  $A)$ $3\log x$  $B)$ $\log x$  $C)$ $6\log x$  $D)$ $5\log x$

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Hint: First we have to define what the terms we need to solve the problem are.
These questions are based on the concept of the logarithm. A log function is defined as $f(x) = {\log _b}x$ where log base is b since base e and $10$ are commonly used and also the division term log is bottom to top of the multiplication.

Since the given question is of multiplication on $x$ and $y$ terms so we separate the x terms on left hand side and then separate the right-hand side in the right-hand side so that we simply apply the log formula and find the value;
Thus, ${x^{2a - 3}}{y^{2a}} = {x^{6 - a}}{y^{5a}} \Rightarrow \dfrac{{{x^{2a - 3}}}}{{{x^{6 - a}}}} = \dfrac{{{y^{5a}}}}{{{y^{2a}}}}$ which is we separated the same terms on same side;
Now turn the denominator into numerator which yields using the property of division ${x^{2a - 3}}{y^{2a}} = {x^{6 - a}}{y^{5a}} \Rightarrow \dfrac{{{x^{2a - 3}}}}{{{x^{6 - a}}}} = \dfrac{{{y^{5a}}}}{{{y^{2a}}}} \Rightarrow {x^{2a - 3 - (6 - a)}} = {y^{5a - 2a}}$ (turning the division terms as multiplication)
${x^{2a - 3 - (6 - a)}} = {y^{5a - 2a}} \Rightarrow 2a - 3 - (6 - a) \times \log x = (5a - 2a) \times \log y$ (On turning the log, the power part will become to the multiplication by the property of the log)
Hence further solving we get and also $2a - 3 - (6 - a) \times \log x = (5a - 2a) \times \log y \Rightarrow (3a - 9)\log x = (5a - 2a) \times \log y$
Thus, after simplifying we get $\log \dfrac{x}{y} = \dfrac{3}{a}\log x$ since we are going to cross multiply the left- and right-hand side; $\log \dfrac{x}{y} = \dfrac{3}{a}\log x \Rightarrow a\log \dfrac{x}{y} = 3\log x$ which is the required equation for the logarithm function.
Note: $\log {x^a} = a\log x$ and $\dfrac{{{x^2}}}{{{x^1}}} = {x^{2 - 1}} = {x^1}$ (Turning the division terms as multiplication) these are some properties of logarithm that we used for this particular given problem, and for separate variable like $x$ and $y$ .