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Logarithms

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Question : 81 of 85

Marks: +1, -0

If

\log_{9} x + \log_{3} x = \frac{3}{2}

x

is

0
1
$\frac{9}{4}$
3

Solution:

Concept:

Use change of base to express both logarithms in the same base.

Explanation:

Step 1: Since

9 = 3^2

, we have

\log_{9} x = \frac{\log_{3} x}{\log_{3} 9} = \frac{\log_{3} x}{2}

.

Step 2: Substitute into the equation:

\frac{\log_{3} x}{2} + \log_{3} x = \frac{3}{2}

.

Step 3: Combine terms:

\frac{3}{2} \log_{3} x = \frac{3}{2}

.

Step 4: Divide both sides by

\frac{3}{2}

:

\log_{3} x = 1

.

Step 5: Therefore,

x = 3^1 = 3

.

Answer:

x = 3

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