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Given that log 2 = 0.3010 and log 3 = 0.4771, find the value of log(1.5)".

Mathematics

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Answer

A. 0.3522\text{A. } 0.3522

Step-by-step solution

5 steps

To find the value of log⁡(1.5)2\log(1.5)^2, we will use the given values log⁡2=0.3010\log 2 = 0.3010 and log⁡3=0.4771\log 3 = 0.4771 and logarithm properties.

  1. Rewrite 1.5 as a fraction.

    1.5=1510=321.5 = \frac{15}{10} = \frac{3}{2}

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

    log⁡(1.5)2=2log⁡(1.5)\log(1.5)^2 = 2 \log(1.5)

  3. Substitute the fractional form of 1.5 and apply the logarithm quotient rule, log⁡(ab)=log⁡a−log⁡b\log\left(\frac{a}{b}\right) = \log a - \log b.

    2log⁡(1.5)=2log⁡(32)=2(log⁡3−log⁡2)2 \log(1.5) = 2 \log\left(\frac{3}{2}\right) = 2 (\log 3 - \log 2)

  4. Substitute the given values for log⁡3\log 3 and log⁡2\log 2.

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

  5. Perform the subtraction and multiplication.

    2(0.1761)=0.35222 (0.1761) = 0.3522

    The value of log⁡(1.5)2\log(1.5)^2 is 0.3522. This matches option A.

    The final answer is A. 0.3522\text{A. } 0.3522.

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