Lesson 6 of 9, 11 min read
Logarithmic functions and their graphs
The logarithmic function as the inverse of an exponential function, its graph, domain, range and vertical asymptote, transformations and finding domains.
The inverse of an exponential
The graph of an inverse function is the reflection of the original graph across the line \(y = x\). Swap the coordinates of the points on \(y = 3^x\) to get points on \(y = \log_3 x\):
| \(y = 3^x\) | \((-1, \frac{1}{3})\) | \((0, 1)\) | \((1, 3)\) | \((2, 9)\) |
|---|---|---|---|---|
| \(y = \log_3 x\) | \((\frac{1}{3}, -1)\) | \((1, 0)\) | \((3, 1)\) | \((9, 2)\) |
Key features
The domain and range are swapped compared with the exponential function, and so are the asymptotes: horizontal for \(b^x\), vertical for \(\log_b x\).
With \(b > 1\), the log is negative for \(0 < x < 1\) and positive for \(x > 1\). It keeps rising forever but very slowly: \(\log x\) only reaches \(6\) when \(x\) is a million.
Transformations
\(y = \log_b(x - h) + k\) shifts the graph \(h\) units horizontally and \(k\) units vertically. The vertical asymptote moves to \(x = h\).
Finding domains
Set every log argument greater than zero and solve.
Practice
Quick check
What is the \(x\)-intercept of every graph \(y = \log_b x\)?
\(\log_b 1 = 0\) for every base.
Quick check
What is the range of \(y = \log x\)?
The domain is restricted to \(x > 0\), but the outputs can be any real number.
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