Question:medium

If 5 - log10 root 1 + x + 4 log10 root 1-x = log10 1/ 1-x2, then 100 x equals

Updated On: Jan 15, 2026
Show Solution

Correct Answer: 99

Solution and Explanation

The provided logarithmic equation is:

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

This equation is simplified through a series of steps.

Step 1: Rewriting the Equation

The initial equation is rewritten as:

\[ 5 - \log_{10} \left( \sqrt{1+x} + 4 \log_{10} \sqrt{1-x} \right) = \log_{10} \left( \frac{1}{\sqrt{1-x^2}} \cdot \frac{1}{\sqrt{1-x}} \right) \]

Further simplification involves factoring and solving.

Step 2: Applying Logarithmic Properties

The left-hand side of the equation is simplified using logarithmic properties:

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

Step 3: Further Simplification

Logarithmic rules are applied to combine terms:

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

This simplifies to:

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

Step 4: Solving for \( x \)

The equation is solved for \( x \):

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

Squaring both sides eliminates the square root:

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

Solving for \( x \) yields:

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

Conclusion

The value of \( x \) is \( \frac{99}{100} \), and the solution is 99/100.

Was this answer helpful?
0

Top Questions on Logarithms

Questions Asked in CAT exam

The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
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.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.
View Solution