Answers


    1. \( \log_5 (125)=3 \)
    2. \( \log_4 (4096)=6 \)
    3. \( \log_{27} \left( 9\sqrt{3}\right)=\frac{5}{6} \)
    4. \( \log_2 \left(\frac{1}{64}\right)=-6 \)
    1. \( 6^3=216 \)
    2. \( 10^3=1000 \)
    3. \( 25^{-2}=\frac{1}{625} \)
    4. \( 16^{\frac{5}{8}}=4\sqrt{2} \)
    1. \( 6 \)
    2. \( 4 \)
    3. \( \frac{1}{2} \)
    4. \( 0 \)
    5. \( -2 \)
    6. \( -3 \)
    7. \( \frac{5}{2} \)
    8. \( \frac{3}{2} \)
    9. \( \frac{5}{3} \)
    10. \( -\frac{3}{2} \)
    11. \( -\frac{3}{2} \)
    12. \( 8 \)
    13. \( \frac{5}{6} \)
    14. \( -4 \)
    15. \( -\frac{3}{2} \)
    1. \( 3.24 \)
    2. \( 7.97 \)
    1. Domain: \( \{x \mid x\in\mathbb{R}\} \), range: \( \{y \mid y \gt 0, y\in\mathbb{R}\} \) The graph of f(x)=4^x, an increasing function with horizonal asymptote at 0 and y-intercept at 1
    2. The function \( g(x) = \log_4 (x) \) is the inverse of the function \( f(x)=4^x \). The graph of \( g(x)=\log_4 (x) \) can be obtained by interchanging the \( x \)- and \( y \)-coordinates of the points on \( f(x)=4^x \) and plotting the new points. The graph of \( g(x)=\log_4 (x) \) can also be obtained by reflecting the graph of \( f(x)=4^x \) in the line \( y=x \).
    3. Domain: \( \{x \mid x \gt 0, x\in\mathbb{R}\} \), range: \( \{y \mid y\in\mathbb{R}\} \) The graphs of f(x)=4^x, g(x)=log_4(x), and y=x showing the reflection of f(x) onto g(x) through y=x

    1. The graphs of A(x)=log_(1/5)(x), B(x)=log_(1/2)(x), C(x)=log_3(x), and D(x)=log_8(x)
    2. All four functions have a domain \( \{x\in\mathbb{R} \mid x \gt 0\} \), a range \( \{y\in\mathbb{R}\} \), an \( x \)-intercept at \( (1,0) \), and a vertical asymptote of \( x=0 \). \( A(x) \) and \( B(x) \) are decreasing functions and \( C(x) \) and \( D(x) \) are increasing functions.
    3. The graph of \( f(x)=\log_\frac{2}{3} (x) \) will lie above \( B(x)=\log_\frac{1}{2} (x) \) when \( 0\lt x \lt1 \) and below \( B(x)=\log_\frac{1}{2} (x) \) when \( x \gt 1 \), intersecting all four functions at \( (1,0) \). The graph of \( g(x)=\log_5 (x) \) will lie between the graphs of the functions \( C(x)=\log_3 (x) \) and \( D(x)=\log_8 (x) \), intersecting all four functions at \( (1,0) \).
      The graphs of f(x)=log_(2/3)(x) and g(x)=log_(1/2)(x) in relation to A(x), B(x), C(x), and D(x)
    1. Vertical translation \( 4 \) units down, domain: \( \{x \mid x \gt 0, x\in\mathbb{R}\} \), range: \( \{y \mid y\in\mathbb{R}\} \), vertical asymptote: \( x=0 \)
      The graph of log_2(x)-4 an increasing function with vertial asymptote at x=0 starting in quadrant IV
    2. Reflection in the \( x \)-axis and horizontal translation left \( 3 \) units, domain: \( \{x \mid x \gt -3, x\in\mathbb{R}\} \), range: \( \{y \mid y\in\mathbb{R}\} \), vertical asymptote: \( x=-3 \)
      The graph of -log_2(x+3), a decreasing function with vertical asymptote at x=-3 and x-intercept at 2
    3. Reflection in the \( y \)-axis and a horizontal stretch about the \( y \)-axis by a factor of \( \frac{1}{2} \), domain: \( \{x \mid x \lt 0, x\in\mathbb{R}\} \), range: \( \{y \mid y\in\mathbb{R}\} \), vertical asymptote: \( x=0 \)
      The graph of log_2(-2x) a decreasing function with vertical asymptote at x=0 starting in quadrant II
    4. Vertical stretch about the \( x \)-axis by a factor of \( 2 \) and vertical translation \( 3 \) units down, domain: \( \{x \mid x \gt 0, x\in\mathbb{R}\} \), range: \( \{y \mid y\in\mathbb{R}\} \), vertical asymptote: \( x=0 \)
      The graph of 2log_2(x-3) an increasing function with vertical asymptote x=0 starting in quadrant IV
    5. Reflection in the \( x \)-axis, vertical stretch about the \( x \)-axis by a factor of \( 3 \), horizontal translation \( 2 \) units right and vertical translation \( 1 \) unit up, domain: \( \{x \mid x \gt 2, x\in\mathbb{R}\} \), range: \( \{y \mid y\in\mathbb{R}\} \), vertical asymptote: \( x=2 \)
      The graph of -3log_2(x-2)+1 a decreasing function with vertical asymptote x=2 starting in quadrant I
    1. \( g^{-1}(x)=\frac{1}{2} \log_3(x-5)+1 \)
    2. \( g^{-1}(x)=5^{\frac{1}{2}\left(x+1\right)}-4 \)
    3. \( g^{-1}(x)=2\left(3^{4-x}\right)+6 \)
    1. The graphs are identical; therefore, we can conclude that \( \log_2(2x)=\log_2(x)+1 \) for \( x \gt 0, x\in\mathbb{R} \).
      The graph of y=log_2(2x)=log_2(x)+1, an increasing function with vertical asymptote at x=0
    2. The graphs are identical; therefore, we can conclude that \( \log_3\left(\frac{x}{3}\right)=\log_3(x)-1 \) for \( x \gt 0, x\in\mathbb{R} \).
      The graph of y=log_3(x/3)=log_3(x)-1, an increasing function with vertical asymptote at x=0
  10. \( (4,2) \) The graphs of 2log_2(x-2) and 2-log_3(x-3) and their intersection at (4,2)