We want to find a possible equation of a logarithmic function from its graph. A general logarithmic function can be written as:
\[ f(x) = a \log_b (x - c) + d \]Before working through the examples below, it is recommended to review the tutorial on graphing logarithmic functions.
Important properties of logarithmic functions include:
For the function \( f(x) = a \log_b (x - c) + d \):
Domain:
\[ x - c > 0 \]Range:
\[ (-\infty, \infty) \]Vertical asymptote:
\[ x - c = 0 \quad \Rightarrow \quad x = c \]Recall the relationship between logarithmic and exponential functions:
\[ y = b^x \quad \Longleftrightarrow \quad x = \log_b y \]Find the logarithmic function of the form \( y = \log_b(x) \) whose graph is shown below.
The point \( (3,1) \) lies on the graph. Substitute into \( y = \log_b(x) \):
\[ 1 = \log_b(3) \]Rewrite in exponential form:
\[ b^1 = 3 \quad \Rightarrow \quad b = 3 \]The logarithmic function is:
\[ y = \log_3(x) \]Find the logarithmic function of the form \( y = \log_b(x - c) \).
From the graph:
\[ (-1,0), \quad (2,2) \]Substitute into \( y = \log_b(x - c) \):
\[ 0 = \log_b(-1 - c) \] \[ 2 = \log_b(2 - c) \]Convert the first equation to exponential form:
\[ b^0 = -1 - c \Rightarrow 1 = -1 - c \Rightarrow c = -2 \]Substitute \( c = -2 \) into the second equation:
\[ 2 = \log_b(4) \]Convert to exponential form:
\[ b^2 = 4 \Rightarrow b = 2 \]The function is:
\[ y = \log_2(x + 2) \]Find the logarithmic function of the form \( y = a \log_4(x - c) \).
The points \( (-1,0) \) and \( (2,3) \) lie on the graph.
\[ a \log_4(-1 - c) = 0 \] \[ a \log_4(2 - c) = 3 \]Divide the first equation by \( a \):
\[ \log_4(-1 - c) = 0 \]Convert to exponential form:
\[ 4^0 = -1 - c \Rightarrow c = -2 \]Substitute into the second equation:
\[ a \log_4(4) = 3 \] \[ a = 3 \]The function is:
\[ y = 3 \log_4(x + 2) \]Find the logarithmic function of the form \( y = a \log_2(x - c) + d \) with vertical asymptote \( x = 1 \).
The asymptote is \( x = c \), so:
\[ c = 1 \]Using points \( (5,0) \) and \( (9,2) \):
\[ 0 = a \log_2(4) + d \] \[ 2 = a \log_2(8) + d \]Simplify:
\[ 0 = 2a + d \] \[ 2 = 3a + d \]Solving:
\[ a = 2, \quad d = -4 \]Final equation:
\[ y = 2 \log_2(x - 1) - 4 \]Find the logarithmic function of the form \( y = a \log_2(x - c) + d \) with vertical asymptote \( x = 0 \).
Solving:
\[ a = -1, \quad d = -4 \]Final equation:
\[ y = -\log_2(x) - 4 \]