## Section 7.3 Logarithmic Functions and Their Graphs

¶The inverse of a simple exponential function, \(f(x)=b^x, b \gt 0, b \neq 1\text{,}\) is called a logarithmic function. The inverse of \(f(x)=b^x\) is symbolized as

which is read aloud as "f-inverse of x equals the log-base-b of x."

Let's consider the function \(f(x)=2^x\) and its inverse \(f^{-1}(x)=\log_2(x)\text{.}\) Recall that if \((a,b)\) is an ordered pair on \(f\text{,}\) then \((b,a)\) is an ordered pair on \(f^{-1}\text{.}\) Several function values for \(f\) are shown in Table 7.3.1 and several function values for \(f^{-1}\) are shown in Table 7.3.2

\(x\) | \(y\) |

\(-3\) | \(0.125\) |

\(-2\) | \(0.25\) |

\(-1\) | \(0.5\) |

\(0\) | \(1\) |

\(1\) | \(2\) |

\(2\) | \(4\) |

\(3\) | \(8\) |

\(4\) | \(16\) |

\(x\) | \(y\) |

\(0.125\) | \(-3\) |

\(0.25\) | \(-2\) |

\(0.5\) | \(-1\) |

\(1\) | \(0\) |

\(2\) | \(1\) |

\(4\) | \(2\) |

\(8\) | \(3\) |

\(16\) | \(4\) |

Let's observe that every ordered pair in Table 7.3.2 satisfies the \(2^y=x\text{.}\) It follows that

This relationship is explored in depth in the section about logarithms as exponents. In the current context, the relationship is a direct result of the inverse function relationship between exponential and logarithmic functions. For the function \(f(x)=2^x\text{,}\) the ordered pairs satisfy the equation \(y=2^x\text{.}\) It immediately follows that the ordered pairs on \(f^{-1}\) satisfy the equation \(2^y=x\text{.}\)

Plots of both \(f(x)=2^x\) and \(f^{-1}(x)=\log_2(x)\) are plotted in Figure 7.3.3 . The line \(y=x\) is also plotted to highlight the fact that the two functions are reflections of one another across that line. Let's first notice that the range of \(f(x)=2^x\text{,}\) \((0,\infty)\text{,}\) becomes the domain of \(f^{-1}(x)=\log_2(x)\text{.}\) Similarly, the domain of \(f(x)=2^x\text{,}\) \((-\infty,\infty)\text{,}\) becomes the range of \(f^{-1}(x)=\log_2(x)\text{.}\)

Additional properties of \(f(x)=2^x\) and their attendant properties of \(f^{-1}(x)=\log_2(x)\) are shown below. Please note that these properties are true whenever the base, \(b\text{,}\) is greater than 1. Note that the symbols \(x \rightarrow 0^+\) are read as "x approaches zero *from the right*."

Properties of \(f(x)=2^x\)

The domain is \((-\infty,\infty)\text{.}\)

The range is \((0,\infty)\text{.}\)

- \(f(0)=1\)
As \(x \rightarrow -\infty\text{,}\) \(y \rightarrow 0\)

As \(x \rightarrow \infty\text{,}\) \(y \rightarrow \infty\)

Properties of \(f^{-1}(x)=\log_2(x)\)

The range is \((-\infty,\infty)\text{.}\)

The domain is \((0,\infty)\text{.}\)

- \(f^{-1}(1)=0\)
As \(x \rightarrow 0^+\text{,}\) \(y \rightarrow -\infty\)

As \(x \rightarrow \infty\text{,}\) \(y \rightarrow \infty\)

Plots of both \(f(x)=0.5^x\) and \(f^{-1}(x)=\log_{0.5}(x)\) are plotted in Figure 7.3.4 . These shapes are representative of \(y=b^x\) and \(y=\log_b(x)\) whenever \(0 \lt b \lt 1\text{.}\) The properties are summarized below.

Properties of \(f(x)=0.5^x\)

The domain is \((-\infty,\infty)\text{.}\)

The range is \((0,\infty)\text{.}\)

- \(f(0)=1\)
As \(x \rightarrow -\infty\text{,}\) \(y \rightarrow \infty\)

As \(x \rightarrow \infty\text{,}\) \(y \rightarrow 0\)

Properties of \(f^{-1}(x)=\log_{0.5}(x)\)

The range is \((-\infty,\infty)\text{.}\)

The domain is \((0,\infty)\text{.}\)

- \(f^{-1}(1)=0\)
As \(x \rightarrow \infty\text{,}\) \(y \rightarrow -\infty\)

As \(x \rightarrow 0^+\text{,}\) \(y \rightarrow \infty\)

One closing comment. In the section about exponential functions and their graphs, it was discussed that on a graph of \(f(x)=2^x\text{,}\) with a scale of inches on each axis, before \(x\) even gets to three feet, the value of \(y\) has already passed the moon. That means that on a similarly scaled graph of \(f^{-1}(x)=\log_2(x)\text{,}\) \(x\) will have to extend beyond the moon before the corresponding value of \(y\) will even get three feet off the ground! So while the range of \(y=\log_2(x)\) is \((-\infty,\infty)\text{,}\) the spectacularly slow pace at which the function increases as \(x \rightarrow \infty\) makes the upper "limit" of the range somewhat misleading.