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Logarithms

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Question : 52 of 85

Marks: +1, -0

If

x^{2 a-3} y^{2 a}=x^{6-a} y^{5 a},

then the value of

a \log \left( \; \frac{x}{y} \right)

is:

$3 \log x$
$\log x$
$6 \log x$
$5 \log x$ .

Solution:

Concept:

The equation involves exponents; take logarithms to separate variables.

Explanation:

Given:

x^{2a-3} y^{2a}=x^{6-a} y^{5a}

.

Take

\log

both sides:

(2a-3)\log x + 2a\log y = (6-a)\log x + 5a\log y

.

Rearrange:

(2a-3 - 6 + a)\log x + (2a - 5a)\log y = 0

.

Simplify:

(3a-9)\log x - 3a\log y = 0

.

Divide by 3:

(a-3)\log x - a\log y = 0

.

Thus,

a\log y = (a-3)\log x

.

Now compute

a\log\left(\frac{x}{y}\right) = a(\log x - \log y) = a\log x - a\log y

.

Substitute

a\log y

:

= a\log x - (a-3)\log x = (a - a + 3)\log x = 3\log x

.

Answer:

3\log x

, which corresponds to option A.

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