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Logarithms

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Question : 44 of 85

Marks: +1, -0

If

3+\log_{5} x=2 \log_{25} y,

x=

$\frac{y}{125}$
$\frac{y}{25}$
$\frac{y^{2}}{625}$
$3-\frac{y^{2}}{25}$

Solution:

Concept:

Use change of base and logarithm properties to simplify the equation.

Explanation:

Given:

3 + \log_{5} x = 2 \log_{25} y

.

Note

\log_{25} y = \frac{\log_{5} y}{\log_{5} 25} = \frac{\log_{5} y}{2}

.

Thus

2 \log_{25} y = 2 \cdot \frac{\log_{5} y}{2} = \log_{5} y

.

So the equation becomes

3 + \log_{5} x = \log_{5} y

.

Then

\log_{5} x = \log_{5} y - 3

.

Write

3 = \log_{5} 125

because

5^3 = 125

.

So

\log_{5} x = \log_{5} y - \log_{5} 125 = \log_{5} \left(\frac{y}{125}\right)

.

Hence

x = \frac{y}{125}

.

Answer:

x = \frac{y}{125}

, which corresponds to option A.

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