Answer :

Let's solve each part of the problem step-by-step.

(a) Solve for [tex]x[/tex] in the equation:

[tex]200(1.1)^x = 20000[/tex]

  1. Divide both sides by 200 to simplify the equation:

    [tex](1.1)^x = \frac{20000}{200}[/tex]

    [tex](1.1)^x = 100[/tex]

  2. Apply logarithms to both sides to solve for [tex]x[/tex]:

    [tex]\log((1.1)^x) = \log(100)[/tex]

  3. Use the logarithmic identity: [tex]\log(a^b) = b \cdot \log(a)[/tex]

    [tex]x \cdot \log(1.1) = \log(100)[/tex]

  4. Solve for [tex]x[/tex]:

    [tex]x = \frac{\log(100)}{\log(1.1)}[/tex]

  5. Calculate the logarithms:

    [tex]x = \frac{2}{0.0414}[/tex] (using [tex]\log_{10}[/tex] approximately)

    [tex]x \approx 48.31[/tex]

Thus, [tex]x \approx 48.31[/tex].

(b) Solve for [tex]x[/tex] in the equation:

[tex]5^x = 2(3)^x[/tex]

  1. Rewrite the right side by separating the terms:

    [tex]5^x = 2 \cdot 3^x[/tex]

  2. Divide both sides by [tex]3^x[/tex]:

    [tex]\left( \frac{5}{3} \right)^x = 2[/tex]

  3. Apply logarithms to both sides:

    [tex]\log((\frac{5}{3})^x) = \log(2)[/tex]

  4. Use the logarithmic identity:

    [tex]x \cdot \log(\frac{5}{3}) = \log(2)[/tex]

  5. Solve for [tex]x[/tex]:

    [tex]x = \frac{\log(2)}{\log(\frac{5}{3})}[/tex]

  6. Calculate the logarithms:

    [tex]x \approx \frac{0.3010}{0.2220}[/tex] (using [tex]\log_{10}[/tex] approximately)

    [tex]x \approx 1.356[/tex]

Therefore, [tex]x \approx 1.356[/tex].

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