Question

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

Answer 1

Correct Answer: 99

We are given the following logarithmic equation:

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

Now, we simplify this equation step by step.

Step 1: First Rewriting

Rewriting the original equation:

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

Next, we simplify further by factoring and solving the equation.

Step 2: Applying Logarithmic Properties

We apply properties of logarithms to simplify the left-hand side of the equation:

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

Step 3: Simplification

We then use logarithmic rules to combine terms:

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

Now, simplifying this equation:

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

Step 4: Solving for \( x \)

We now solve for \( x \). Using the equation:

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

Square both sides to get rid of the square root:

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

Solving for \( x \):

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

Conclusion

Therefore, the value of \( x \) is \( 99 \), and the solution is 99.