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Simplify log 6 - log 15 + 2 log 5 and log (5/8) - 2 log (5/9) + log 40

Mathematics

ScanSolve AI Answer

Step-by-step solution

13 steps

1(a)

  1. Start with the given equation.

    log⁡b=log⁡15+2log⁡5\log b = \log 15 + 2 \log 5
  2. Apply the power rule

    nlog⁡a=log⁡(an)n \log a = \log (a^n).

    2log⁡5=log⁡(52)2 \log 5 = \log (5^2) log⁡b=log⁡15+log⁡(52)\log b = \log 15 + \log (5^2)
  3. Apply the product rule

    log⁡a+log⁡c=log⁡(a⋅c)\log a + \log c = \log (a \cdot c).

    log⁡b=log⁡(15⋅52)\log b = \log (15 \cdot 5^2)
  4. Compute 52=255^2 = 25.

    log⁡b=log⁡(15⋅25)\log b = \log (15 \cdot 25)
  5. Compute 15⋅25=37515 \cdot 25 = 375.

    log⁡b=log⁡375\log b = \log 375
  6. If log⁡b=log⁡375\log b = \log 375, then b=375b = 375.

    Final answer: b=375b = 375
    375

    2)

  7. Start with the given expression.

    log⁡53−2log⁡37+log⁡44\log \frac{5}{3} - 2 \log \frac{3}{7} + \log 44
  8. Apply the power rule to the second term

    nlog⁡a=log⁡(an)n \log a = \log (a^n).

    −2log⁡37=−log⁡(37)2=−log⁡949-2 \log \frac{3}{7} = -\log \left( \frac{3}{7} \right)^2 = -\log \frac{9}{49}
  9. Apply the quotient rule to the negative log

    −log⁡949=log⁡499-\log \frac{9}{49} = \log \frac{49}{9}.

    log⁡53+log⁡499+log⁡44\log \frac{5}{3} + \log \frac{49}{9} + \log 44
  10. Apply the product rule for logs

    log⁡a+log⁡b+log⁡c=log⁡(a⋅b⋅c)\log a + \log b + \log c = \log (a \cdot b \cdot c).

    log⁡(53⋅499⋅44)\log \left( \frac{5}{3} \cdot \frac{49}{9} \cdot 44 \right)
  11. Simplify the argument step-by-step.

    First, 53⋅499\frac{5}{3} \cdot \frac{49}{9}.
    Numerator: 5⋅49=2455 \cdot 49 = 245
    Denominator: 3⋅9=273 \cdot 9 = 27

    24527⋅44\frac{245}{27} \cdot 44
  12. Now multiply by 44.

    245⋅44=10780245 \cdot 44 = 10780 log⁡1078027\log \frac{10780}{27}
  13. Check if 1078027\frac{10780}{27} simplifies (gcd of 10780 and 27 is 1).

    Final answer: log⁡1078027\log \frac{10780}{27}
    \log \dfrac{10780}{27}

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