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If \(5 - log_{10}\  \sqrt {1 + x }+ 4\  log_{10 }\  \sqrt {1-x} = log_{10}\  \frac {1}{\sqrt {1-x^2}}\), then \(100 x \) equals

Updated On: Jan 15, 2026
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Solution and Explanation

The given equation is:

\[ 5 - \log_{10} \sqrt{1+x} + 4 \log_{10} \sqrt{1-x} = \log_{10} \frac{1}{\sqrt{1-x^2}} \]

Step 1: Equation Rearrangement

The right-hand side is rewritten as:

\[ 5 - \log_{10} \sqrt{1+x} + 4 \log_{10} \sqrt{1-x} = \log_{10} \left( \sqrt{1+x} \times \sqrt{1-x} \right)^{-1} \]

Step 2: Applying Logarithm Properties

Using the property \( \log_b(a^n) = n \log_b a \):

\[ 5 - \log_{10} \sqrt{1+x} + 4 \log_{10} \sqrt{1-x} = (-1) \log_{10} \left( \sqrt{1+x} \right) + (-1) \log_{10} \left( \sqrt{1-x} \right) \]

Step 3: Term Consolidation

The equation simplifies to:

\[ 5 = -\log_{10} \sqrt{1+x} + \log_{10} \sqrt{1+x} - \log_{10} \sqrt{1-x} - 4 \log_{10} \sqrt{1-x} \]

Step 4: Further Simplification

The equation reduces to:

\[ 5 = -5 \log_{10} \sqrt{1-x} \]

Step 5: Solving for \( x \)

Isolating \( \sqrt{1-x} \):

\[ \sqrt{1-x} = \frac{1}{10} \]

Squaring both sides yields:

\[ (\sqrt{1-x})^2 = \frac{1}{100} \]

This gives:

\[ 1 - x = \frac{1}{100} \]

Therefore:

\[ x = 1 - \frac{1}{100} = \frac{99}{100} \]

Step 6: Calculating the Final Result

The value of \( 100x \) is:

\[ 100x = 100 \times \frac{99}{100} = 99 \]

Conclusion:

The final answer is \( \boxed{99} \).

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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
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