Question

The approximate value of \( \log_{10}99 \) is (Given \( \log_{10}e = 0.4343 \))

• A. 1.9657
• B. 1.9857
• C. 1.9957
• D. 1.9757

Hint:

For numbers close to powers of 10, use logarithmic approximation formulas to quickly estimate values.

Answer 1

The Correct Option is C

Step 1: Express 99 suitably.
\[ \log_{10}99 = \log_{10}(100 - 1) \] Step 2: Use logarithmic approximation.
For small \(x\), \[ \log(1-x) \approx -x \log e \] Thus, \[ \log_{10}99 = \log_{10}100 + \log_{10}\left(1-\frac{1}{100}\right) \] Step 3: Substitute values.
\[ = 2 - \frac{1}{100}\log_{10}e \] \[ = 2 - \frac{1}{100}(0.4343) \] Step 4: Final calculation.
\[ \log_{10}99 \approx 2 - 0.004343 = 1.9957 \]