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Log_5 81 + log_5 3 / log_5 24³ - log_5 9

Mathematics

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Answer

5log⁡53log⁡51536\frac{5 \log_5 3}{\log_5 1536}

Step-by-step solution

3 steps

Here's how to solve the expression:

log⁡581+log⁡53log⁡5243−log⁡59\frac{\log_5 81 + \log_5 3}{\log_5 24^3 - \log_5 9}

  1. Simplify the numerator using the logarithm property log⁡bM+log⁡bN=log⁡b(MN)\log_b M + \log_b N = \log_b (MN).

    log⁡581+log⁡53=log⁡5(81×3)\log_5 81 + \log_5 3 = \log_5 (81 \times 3) =log⁡5243= \log_5 243

    Since 243=35243 = 3^5, we can write:

    =log⁡535=5log⁡53= \log_5 3^5 = 5 \log_5 3

  2. Simplify the denominator using the logarithm property log⁡bM−log⁡bN=log⁡b(MN)\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right).

    log⁡5243−log⁡59=log⁡5(2439)\log_5 24^3 - \log_5 9 = \log_5 \left(\frac{24^3}{9}\right)

    First, calculate 24324^3:

    243=24×24×24=576×24=1382424^3 = 24 \times 24 \times 24 = 576 \times 24 = 13824

    Now substitute this back into the expression:

    =log⁡5(138249)= \log_5 \left(\frac{13824}{9}\right) =log⁡51536= \log_5 1536

  3. Combine the simplified numerator and denominator to get the final expression.

    5log⁡53log⁡51536\frac{5 \log_5 3}{\log_5 1536}
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