Answers


    1. \( 5 \)
    2. \( -3 \)
    3. \( 0 \)
    4. \( 1 \)
    5. \( -1 \)
    6. \( 81 \)
    7. \( 0.0001 \)
    8. \( 12 \)
    9. \( \frac{1}{2} \)
    1. no restrictions on \(x\), \(x=-\frac{3}{2}\)
    2. \(x \gt -\frac{3}{2}, x=\frac{3}{4}\)
    3. \(x \gt \frac{2}{3}, x=4\)
    4. \(x \gt 0, x=\frac{1}{4}\)
    5. \(x \gt 0, x=16\)
    6. \(0\lt x \lt 6, x=2\)
    1. \(\log_2 \left(\dfrac{rs}{t}\right), \ r,s,t \gt 0\)
    2. \(\log \left(\dfrac{a^3}{9b^2}\right), \ a,b \gt0\)
    3. \(\log_y (x),\ x,y \gt 0\)
    4. \(\log_5\left(\dfrac{2\sqrt{x}}{x-3}\right), \ x \gt 3 \)
    5. \(\log\left(\sqrt[4]{(y-1)(x+1)}\right), \ x \gt -1, y \gt 1\)
    6. \(\log_3\left(\dfrac{a^3}{b\sqrt{c}}\right), \ a,b,c \gt 0\)
    1. \( 2 \)
    2. \( 3 \)
    3. \( -2 \)
    4. \( 36 \)
    5. \( 2 \)
    6. \( \frac{5}{2} \)
    7. \( 3 \)
    8. \( 16 \)
    9. \( 6 \)
    1. \( 3 + \log_2 (a) \)
    2. \( 2\log_2 (a) + \log_2 (b) \)
    3. \( \frac{1}{2}\log_2 (a) - \log_2 (b) -log_2 (c) \)
    4. \( -5+5\log_2 (a) + 5\log_2 (b) \)
    5. \( 2\log_2 (a) +\frac{1}{2} \log_2 (b) - 3\log_2 (c) \)
    6. \( \frac{1}{4}\left(1-\log_2(b)-\log_2(c)\right) \)
    1. \( 2.267 \)
    2. \( 3.800 \)
    3. \( 0.774 \)
    4. \( -4.585 \)
    1. \( x \approx 1.700 \)
    2. \( x \approx 3.236 \)
    3. \( x \approx -1.661 \)
    4. \( x \approx 7.209 \)
    1. Proof; see solutions.
    2. Proof; see solutions.
    1. \(f(x)=\log_2 \left(4x^3\right)=\log_2 (4)+ 3\log_2 (x)=2+3\log_2 (x)\) The graph of log_2(8x^3), an increasing function with vertical asymptote at x=0
      The graph of \(f(x)=\log_2 \left(4x^3\right)\) and the graph of \(f(x)=2+3\log_2 (x)\) are identical.
    2. The graph of \(g(x)=\log_2 \left(x^2\right)\) is not the same as the graph of \(h(x)=2\log_2 (x)\). The graph of \(g(x)=\log_2 \left(x^2\right)\) is composed of two branches defined by \(h(x)=2\log_2 (x)\) and \(h(x)=2\log_2 (-x)\).
      The graph of g(x)=log_2(x^2) a function decreasing until a vertical asymptote at x=0 then increasing
      The graph of h(x)=2log_2(x), an increasing function with vertical asymptote at x=0, x-intercept at 1

    3. The laws of logarithms, in particular the power law \(\log_c \left(x^n\right)=n\log_c (x)\), can only be applied when the base of the argument is positive in value (\(x \gt 0\)).

      The domain of the function \(f(x)=\log_2 \left(4x^3\right)\) is \(\{x \mid x \gt 0, x\in\mathbb{R}\}\). Applying the laws of logarithms within the domain of the function provides us with an equivalent equation for the function, \(f(x)=2+3\log_2 (x)\).

      The domain of \(g(x)=\log_2 \left(x^2\right)\) is \(\{x \mid x\neq0, x\in\mathbb{R}\}\). The power law cannot be applied to this equation within the domain of \(g(x)\). Thus \(g(x)=\log_2 \left(x^2\right)\) and \(h(x)=2\log_2 (x)\) are not equivalent forms of the same function.

      Note: An equivalent equation for the graph of \(g(x)\) would be \(g(x)=2\log_2 \lvert x\rvert\).

  10. Proof; see solutions.