Honors Algebra 2: Extra Logarithms Review Problems

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  1. Simplify: \( \displaystyle \log {\left( \frac{1}{100} \right)} + 8x^{3} - 5e^{3\ln{x}} \)
    2. Solution

    \( \displaystyle 3x^3 - 2 \)

  2. Simplify: \( \displaystyle \log_{\frac{1}{3}} \left( \sqrt[6]{3} \right)^{9\sqrt{3}} \)
    3. Solution

    \( \displaystyle - \frac{3 \sqrt{3}}{2} \)

  3. Solve: \( \displaystyle \left( \log_{3}{x} \right)^2 - \log_{\sqrt{3}} {x} = \log_4 {64} \)
    4. Solution

    \( \displaystyle x = \left\{ \frac{1}{3}, 27 \right\} \)

  4. Solve: \( \displaystyle \log_2 {(3 - x)} - \log_{\sqrt{2}}2 \ge \frac{1}{2}\)
    5. Solution

    \( \displaystyle x \le 3 - 4 \sqrt{2} \)

  5. Solve: \( \displaystyle \log_{\frac{1}{2}} {(x - 5)} + \log_{\frac{1}{2}} {(x + 2)} \le -3\)
    6. Solution

    \( \displaystyle x \in [ 6, \infty ) \)

  6. Solve: \( \displaystyle \log_6 {(15 - x)} \le 2 - \log_6 {x} \)
    7. Solution

    \( \displaystyle x \in (0, 3] \cup [12, 15] \)

  7. Simplify: \( \displaystyle 10^{\frac{1}{2} \log{x}} + \log_n{ (\sqrt{3} )} \cdot \log_9{\left( n^2 \right)} \)
    8. Solution

    \( \displaystyle \sqrt{x} + \frac{1}{2} \)

  8. Let \( \displaystyle \log_7{4} = a \) and \( \displaystyle \log_7{3} = b \). Evaluate each of the following. Answer in terms of a, and b.
    1. \( \displaystyle \log_7{48}. \)
      Solution

      \( \displaystyle 2a + b \)

    2. \( \displaystyle \log_7{\frac{27}{49}}.\)
      3. Solution

      \( \displaystyle 3b - 2 \)

    3. \( \displaystyle \log_7{6} \)
      4. Solution

      \( \displaystyle \frac{a}{2} + 3 \)

  9. You have invested your oney in an account that is compounded monthly at an annua rate of 2.15%.
    1. When will you triple your investment? Answer to the nearest hundredth.
      2. Solution

      5.14 years

    2. Solve: What is the effective annual yield? Answer to the nearest hundredth.
      3. Solution

      2.17 %

  10. Graph: \( \displaystyle y = 4 - \log_2{(x + 3)} \)
    11. Solution

    Check using your graphing calculator or Desmos.

  11. A challenging problem! Given: \( \displaystyle \log_c{2} = w, \log_c{3} = y,\) and \( \displaystyle \log_c{5} = z,\) find \( \displaystyle \log_{\frac{1}{c}}{\sqrt{2.4}}\) in terms of \(x, y, \) and/or \(z\).
    12. Solution

    \( \displaystyle 2 - \frac{y}{2} + \frac{z}{2} \)