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Without using tables, simplify (log 27)(log 36) / (log 216)(log 3)

Mathematics

ScanSolve AI Answer

Answer

2

Step-by-step solution

4 steps

Here's how to simplify the expression:

  1. Express each number inside the logarithm as a power of its prime factors or a common base.

    27=3327 = 3^3 36=6236 = 6^2 216=63216 = 6^3 3=313 = 3^1

  2. Apply the logarithm property log⁡b(xk)=klog⁡bx\log_b (x^k) = k \log_b x to each term.

    log⁡27=log⁡33=3log⁡3\log 27 = \log 3^3 = 3 \log 3 log⁡36=log⁡62=2log⁡6\log 36 = \log 6^2 = 2 \log 6 log⁡216=log⁡63=3log⁡6\log 216 = \log 6^3 = 3 \log 6 log⁡3=log⁡3\log 3 = \log 3

  3. Substitute these simplified logarithmic terms back into the original expression.

    (log⁡27)(log⁡36)(log⁡216)(log⁡3)=(3log⁡3)(2log⁡6)(3log⁡6)(log⁡3)\frac{(\log 27)(\log 36)}{(\log 216)(\log 3)} = \frac{(3 \log 3)(2 \log 6)}{(3 \log 6)(\log 3)}

  4. Cancel out common terms from the numerator and the denominator and simplify the numerical coefficients.

    3×2×(log⁡3)×(log⁡6)3×(log⁡6)×(log⁡3)=6(log⁡3)(log⁡6)3(log⁡3)(log⁡6)\frac{3 \times 2 \times (\log 3) \times (\log 6)}{3 \times (\log 6) \times (\log 3)} = \frac{6 (\log 3)(\log 6)}{3 (\log 3)(\log 6)} 63=2\frac{6}{3} = 2

    The final answer is 2.

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