Logarithm Functions
In the previous chapter, we defined the logarithm as the inverse operation of exponentiation. Just as these two operations are algebraically linked, their graphs are also deeply connected.
This chapter is dedicated to visualizing logarithmic functions. We will see how the graph of a logarithmic function, such as \(y = \log(x)\), is a reflection of its corresponding exponential function, \(y = 10^x\), across the line \(y=x\).
We will explore the key features that define these graphs, including their domain and range, their vertical asymptotes, and their intercepts.
This chapter is dedicated to visualizing logarithmic functions. We will see how the graph of a logarithmic function, such as \(y = \log(x)\), is a reflection of its corresponding exponential function, \(y = 10^x\), across the line \(y=x\).
We will explore the key features that define these graphs, including their domain and range, their vertical asymptotes, and their intercepts.
The Logarithmic Function
Definition Logarithmic Function
The logarithmic function is \(f(x) = \log(x)\). It is the inverse of the exponential function \(g(x)=10^x\).
- Domain: \((0, +\infty)\)
- Range: \((-\infty, +\infty)\)
Proposition Graph of the Logarithmic Function
The graph of \(\textcolor{colordef}{y = \log(x)}\) is the reflection of the graph of \(\textcolor{colorprop}{y = 10^x}\) over the line \(\textcolor{olive}{y = x}\).Key features of the graph of \(y=\log(x)\):
- It has a vertical asymptote at the y-axis (\(x=0\)).
- It passes through the point (1, 0).
- It is an increasing function.