This page provides detailed answers and explanations for the questions in Tutorial (4) on Logarithmic Functions.
Investigate base \( B \): set \( a = 1 \), \( b = 1 \), \( c = 0 \), and \( d = 0 \). Vary \( B \) between values greater than 0 and less than 1, and values greater than 1. Observe the resulting graphs and explain the differences.
For logarithmic functions of the form
\[ f(x) = \log_B(x), \]the behavior depends on the base \( B \):
This occurs because the logarithm is the inverse of the exponential function, whose monotonicity depends on the base.
Investigate the effect of parameter \( a \) (vertical scaling) by setting \( B = e \), \( b = 1 \), \( c = 0 \), and \( d = 0 \).
The function is
\[ f(x) = a \log_B(x). \]As \( a \) increases, the graph is stretched vertically. As \( a \) decreases (but remains nonzero), the graph is compressed vertically. If \( a < 0 \), the graph is also reflected across the \( x \)-axis.
Investigate the effect of parameter \( b \) (horizontal scaling) by setting \( a = 1 \), \( c = 0 \), \( d = 0 \), and \( B = e \).
The function becomes
\[ f(x) = \log_B(bx). \]As \( b \) increases, the graph is horizontally compressed. As \( b \) decreases (but remains positive), the graph is horizontally stretched.
Set \( B = e \), \( a = 1 \), and \( b = 1 \). Investigate the effects of parameters \( c \) (horizontal shift) and \( d \) (vertical translation).
The function is
\[ f(x) = \log_B(x + c) + d. \]If \( c > 0 \), the graph is shifted to the left. If \( c < 0 \), the graph is shifted to the right.
If \( d > 0 \), the graph is shifted upward. If \( d < 0 \), the graph is shifted downward.
Set \( B \), \( a \), and \( d \) to fixed values. Explain how parameters \( b \) and \( c \) affect the domain of the logarithmic function. Provide an analytical explanation.
The function is
\[ f(x) = a \log_B(bx + c) + d. \]The logarithm is defined only when its argument is positive, so the domain satisfies
\[ bx + c > 0. \]Solving this inequality determines the domain of the function.
Which parameter(s) affect the \( x \)-intercept? Is there always an \( x \)-intercept? Explain analytically.
The \( x \)-intercept occurs when \( f(x) = 0 \):
\[ a \log_B(bx + c) + d = 0. \]Solving step by step:
\[ \log_B(bx + c) = -\frac{d}{a} \] \[ bx + c = B^{-\frac{d}{a}} \] \[ x = \frac{B^{-\frac{d}{a}} - c}{b}. \]An \( x \)-intercept exists provided \( a \neq 0 \) and \( b \neq 0 \), and the resulting \( x \) lies in the domain.
Which parameter(s) affect the \( y \)-intercept? Is there always a \( y \)-intercept? Explain analytically.
The \( y \)-intercept occurs at \( x = 0 \):
\[ y = f(0) = a \log_B(c) + d. \]A \( y \)-intercept exists only if
\[ c > 0. \]Which parameter(s) affect the vertical asymptote? Explain analytically.
The vertical asymptote occurs where the argument of the logarithm equals zero:
\[ bx + c = 0. \]Solving for \( x \) gives
\[ x = -\frac{c}{b}. \]Thus, the vertical asymptote depends on parameters \( b \) and \( c \). The graph always has a vertical asymptote provided \( b \neq 0 \).