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Evaluate the following: i) log₅(3/2) + 3 log₅(15/2) - log₅(81/8) ii) 1/3 log₂ 64

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2 parts
i

3−log⁡5(2)3 - \log_{5}(2)

ii
2

Step-by-step solution

7 steps

Here are the evaluations for the given expressions:

Part i· 4 steps

Evaluate log⁡5(32)+3log⁡5(152)−log⁡5(818)\log_{5}\left(\frac{3}{2}\right) + 3\log_{5}\left(\frac{15}{2}\right) - \log_{5}\left(\frac{81}{8}\right)

  1. Apply the power rule of logarithms, nlog⁡bx=log⁡bxnn \log_b x = \log_b x^n, to the second term.

    3log⁡5(152)=log⁡5((152)3)=log⁡5(15323)=log⁡5(33758)3\log_{5}\left(\frac{15}{2}\right) = \log_{5}\left(\left(\frac{15}{2}\right)^3\right) = \log_{5}\left(\frac{15^3}{2^3}\right) = \log_{5}\left(\frac{3375}{8}\right)

    The expression becomes:

    log⁡5(32)+log⁡5(33758)−log⁡5(818)\log_{5}\left(\frac{3}{2}\right) + \log_{5}\left(\frac{3375}{8}\right) - \log_{5}\left(\frac{81}{8}\right)

  2. Combine the terms using the product rule log⁡bx+log⁡by=log⁡b(xy)\log_b x + \log_b y = \log_b (xy) and the quotient rule log⁡bx−log⁡by=log⁡b(xy)\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right).

    log⁡5(32×33758818)\log_{5}\left(\frac{\frac{3}{2} \times \frac{3375}{8}}{\frac{81}{8}}\right)

    Simplify the fraction inside the logarithm:

    32×33758×881\frac{3}{2} \times \frac{3375}{8} \times \frac{8}{81}

    The 8 in the denominator and numerator cancel out:

    3×33752×81\frac{3 \times 3375}{2 \times 81}

  3. Simplify the numerical fraction by factoring.

    3×(33×53)2×34=34×532×34\frac{3 \times (3^3 \times 5^3)}{2 \times 3^4} = \frac{3^4 \times 5^3}{2 \times 3^4}

    Cancel out 343^4:

    532=1252\frac{5^3}{2} = \frac{125}{2}

    So the expression becomes:

    log⁡5(1252)\log_{5}\left(\frac{125}{2}\right)

  4. Apply the quotient rule log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y and evaluate.

    log⁡5(1252)=log⁡5(125)−log⁡5(2)\log_{5}\left(\frac{125}{2}\right) = \log_{5}(125) - \log_{5}(2)

    Since 125=53125 = 5^3, we have log⁡5(125)=3\log_{5}(125) = 3.

    3−log⁡5(2)3 - \log_{5}(2)

    The final answer is 3−log⁡5(2)\boxed{3 - \log_{5}(2)}.

Part ii2

Evaluate 13log⁡264\frac{1}{3}\log_{2} 64

  1. Apply the power rule of logarithms, nlog⁡bx=log⁡bxnn \log_b x = \log_b x^n.

    13log⁡264=log⁡2(641/3)\frac{1}{3}\log_{2} 64 = \log_{2} (64^{1/3})

  2. Evaluate the term inside the logarithm.

    641/3=64364^{1/3} = \sqrt[3]{64}

    Since 4×4×4=644 \times 4 \times 4 = 64, we have 643=4\sqrt[3]{64} = 4.
    The expression becomes:

    log⁡24\log_{2} 4

  3. Evaluate the logarithm.

    We need to find the power to which 2 must be raised to get 4.
    Since 22=42^2 = 4, we have log⁡24=2\log_{2} 4 = 2.
    The final answer is 2\boxed{2}.

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