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12.47 × 0.0725 / [435] ^ (1/5)

Mathematics

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Answer

0.2682

Step-by-step solution

5 steps

To evaluate the expression 12.47×0.07254351/5\frac{12.47 \times 0.0725}{435^{1/5}} using logarithms, let X=12.47×0.07254351/5X = \frac{12.47 \times 0.0725}{435^{1/5}}.

We use the logarithm properties:
log⁡(ab)=log⁡a+log⁡b\log(ab) = \log a + \log b
log⁡(ab)=log⁡a−log⁡b\log(\frac{a}{b}) = \log a - \log b
log⁡(an)=nlog⁡a\log(a^n) = n \log a

So, log⁡X=log⁡(12.47)+log⁡(0.0725)−15log⁡(435)\log X = \log(12.47) + \log(0.0725) - \frac{1}{5}\log(435).

We will use a log table (or calculator for log values) to find the logarithms to 4 decimal places.

  1. Find the logarithms of the individual numbers.

    • log⁡12.47\log 12.47: The characteristic is 1. The mantissa for 12.47 is 0.0959. So, log⁡12.47=1.0959\log 12.47 = 1.0959.
    • log⁡0.0725\log 0.0725: The characteristic is 2ˉ\bar{2} (since the first non-zero digit is in the second decimal place). The mantissa for 7.25 is 0.8603. So, log⁡0.0725=2ˉ.8603\log 0.0725 = \bar{2}.8603.
    • log⁡435\log 435: The characteristic is 2. The mantissa for 4.35 is 0.6385. So, log⁡435=2.6385\log 435 = 2.6385.

  2. Calculate the logarithm of the numerator.

    log⁡(12.47×0.0725)=log⁡12.47+log⁡0.0725\log(12.47 \times 0.0725) = \log 12.47 + \log 0.0725 =1.0959+2ˉ.8603= 1.0959 + \bar{2}.8603 =1.0959+(−2+0.8603)= 1.0959 + (-2 + 0.8603) =(1−2)+(0.0959+0.8603)= (1 - 2) + (0.0959 + 0.8603) =−1+0.9562=1ˉ.9562= -1 + 0.9562 = \bar{1}.9562

  3. Calculate the logarithm of the denominator.

    log⁡(4351/5)=15log⁡435\log(435^{1/5}) = \frac{1}{5} \log 435 =15×2.6385= \frac{1}{5} \times 2.6385 =0.5277= 0.5277

  4. Calculate log⁡X\log X.

    log⁡X=log⁡(numerator)−log⁡(denominator)\log X = \log(\text{numerator}) - \log(\text{denominator}) log⁡X=1ˉ.9562−0.5277\log X = \bar{1}.9562 - 0.5277 =(−1+0.9562)−0.5277= (-1 + 0.9562) - 0.5277 =−1+(0.9562−0.5277)= -1 + (0.9562 - 0.5277) =−1+0.4285=1ˉ.4285= -1 + 0.4285 = \bar{1}.4285

  5. Find the antilogarithm of log⁡X\log X.

    The characteristic is 1ˉ\bar{1}, which means the first significant digit of the number is in the first decimal place.
    The mantissa is 0.4285.
    The antilog of 0.4285 is approximately 2.682.
    Therefore, X=2.682×10−1=0.2682X = 2.682 \times 10^{-1} = 0.2682.

    We can summarize the calculation in a table format:

    NumberLogarithm12.471.09590.07252ˉ.8603Numerator Product1ˉ.95624352.63854351/515×2.6385=0.5277Log X1ˉ.9562−0.5277=1ˉ.4285Antilog(1ˉ.4285)0.2682\begin{array}{|c|c|} \hline \text{Number} & \text{Logarithm} \\ \hline 12.47 & 1.0959 \\ 0.0725 & \bar{2}.8603 \\ \hline \text{Numerator Product} & \bar{1}.9562 \\ \hline 435 & 2.6385 \\ 435^{1/5} & \frac{1}{5} \times 2.6385 = 0.5277 \\ \hline \text{Log X} & \bar{1}.9562 - 0.5277 = \bar{1}.4285 \\ \hline \text{Antilog}(\bar{1}.4285) & 0.2682 \\ \hline \end{array}

    The final answer is 0.2682.

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