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Given that log 10 2 = 0.3010 and log 10 3 = 0.4771, evaluate correct to 2 significant figures and without using tables or calculator, log 10 125.

Mathematics

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Answer

0.051\text{0.051}

Step-by-step solution

10 steps

To evaluate log⁡101.125\log_{10} 1.125 without a calculator, we first convert 1.125 into a fraction and then use the given logarithm values and properties.

  1. Convert the decimal to a fraction.

    1.125=112510001.125 = \frac{1125}{1000}

  2. Simplify the fraction.

    Divide the numerator and denominator by common factors.

    11251000=225200=4540=98\frac{1125}{1000} = \frac{225}{200} = \frac{45}{40} = \frac{9}{8}

  3. Rewrite the logarithm using the simplified fraction.

    log⁡101.125=log⁡10(98)\log_{10} 1.125 = \log_{10} \left(\frac{9}{8}\right)

  4. Apply the logarithm quotient rule, log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b.

    log⁡10(98)=log⁡109−log⁡108\log_{10} \left(\frac{9}{8}\right) = \log_{10} 9 - \log_{10} 8

  5. Express 9 and 8 in terms of 2 and 3.

    9=329 = 3^2 8=238 = 2^3

    Substitute these into the expression:

    log⁡10(32)−log⁡10(23)\log_{10} (3^2) - \log_{10} (2^3)

  6. Apply the logarithm power rule, log⁡(ab)=blog⁡a\log (a^b) = b \log a.

    2log⁡103−3log⁡1022 \log_{10} 3 - 3 \log_{10} 2

  7. Substitute the given values log⁡102=0.3010\log_{10} 2 = 0.3010 and log⁡103=0.4771\log_{10} 3 = 0.4771.

    2(0.4771)−3(0.3010)2(0.4771) - 3(0.3010)

  8. Perform the multiplications.

    0.9542−0.90300.9542 - 0.9030

  9. Perform the subtraction.

    0.05120.0512

  10. Round the result to 2 significant figures.

    The first non-zero digit is 5, and the second is 1. The digit after 1 is 2, which is less than 5, so we round down.

    0.0510.051
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