Introduction
A logarithmic graph describes how the exponent changes as a positive input changes. Its shape depends on the base: some logarithmic functions increase from left to right, while others decrease. In either case, the graph is confined to positive \(x\)-values and approaches the \(y\)-axis without touching it.
We will examine the graph of \(y=\log_b(x)\), where \(b\) is a fixed base. The inverse relationship between logarithms and exponentials gives a direct way to find its domain, range, intercept, asymptote, and direction of change.
The Logarithmic Function and Its Base
A logarithm \(\log_b(x)\) is the exponent to which \(b\) must be raised to obtain \(x\). The base must be positive and cannot equal \(1\). Also, the input \(x\) must be positive.
In other words, \(y=\log_b(x)\) and \(b^y=x\) express the same relationship. The exponential function \(b^y\) is always positive, so no logarithm with base \(b\) can have zero or a negative number as its input. Conversely, every positive input \(x\) has a logarithm, because it can be written as a power of \(b\).
The restriction \(b\ne 1\) is essential. If \(b=1\), then \(1^y=1\) for every real \(y\), so the equation \(1^y=x\) would have no solution for \(x\ne 1\) and would not have a unique solution for \(x=1\). It would not define a logarithmic function on the positive real numbers.
Domain, Range, and Intercept
The logarithm's domain follows from the requirement \(x>0\). Its range follows by considering the output: for every real number \(y\), the value \(x=b^y\) is positive, and the definition then gives \(\log_b(x)=y\). Thus every real number occurs as an output.
The graph crosses the \(x\)-axis when \(y=0\). Since \(b^0=1\), the logarithm of \(1\) is \(0\), so the intercept is \((1,0)\). There is no \(y\)-intercept: the \(y\)-axis has \(x=0\), and \(x=0\) is outside the domain.
The Vertical Asymptote
The graph lies only to the right of the \(y\)-axis. As \(x\) gets closer to zero through positive values, the logarithm's output grows without bound in one direction or the other. This makes the vertical line \(x=0\) a vertical asymptote.
For a logarithmic graph, \(x=0\) is the only vertical asymptote. The direction in which the graph rises or falls near this line depends on whether \(b\) is greater than or less than \(1\). In both cases, the graph does not reach \(x=0\), because \(0\) is not in its domain.
Why the Base Determines Whether the Graph Increases or Decreases
The exponential function \(b^y\) is increasing when \(b>1\) and decreasing when \(0<b<1\). A logarithm reverses the roles of the input and output in the equation \(b^y=x\), but whether its graph increases or decreases still follows from the behavior of the exponential function. Here is a direct argument.
To prove this, take any positive \(x_1<x_2\), and set \(y_1=\log_b(x_1)\) and \(y_2=\log_b(x_2)\). By the definition of logarithm, \(b^{y_1}=x_1\) and \(b^{y_2}=x_2\). Therefore \(b^{y_1}<b^{y_2}\). When \(b>1\), the exponential function is increasing, so this inequality requires \(y_1<y_2\). When \(0<b<1\), the exponential function is decreasing, so the same inequality requires \(y_1>y_2\). These are exactly the definitions of an increasing and a decreasing function, respectively.
The behavior near the asymptote is consistent with this result. If \(b>1\), inputs near zero give very negative logarithms, while large inputs give positive logarithms. If \(0<b<1\), inputs near zero give very positive logarithms, while large inputs give negative logarithms.
Building and Checking Points
To find useful points for a sketch, choose an exponent \(y\), calculate \(x=b^y\), and plot \((x,y)\). This method works because the resulting pair satisfies the defining equation \(b^y=x\), so \(y=\log_b(x)\). It often produces exact points without a calculator.
Worked Example: Graph \(y=\log_2(x)\)
Here the base is \(b=2\), so \(b>1\). The graph is increasing. Its domain is \((0,\infty)\), its range is \(\mathbb{R}\), and its \(x\)-intercept is \((1,0)\). It has no \(y\)-intercept and has vertical asymptote \(x=0\).
To find exact points, choose \(y=-2,-1,0,1,2\) and calculate \(x=2^y\):
The points are \(\left(\frac14,-2\right)\), \(\left(\frac12,-1\right)\), \((1,0)\), \((2,1)\), and \((4,2)\). Check them by substituting each \(y\)-value into \(2^y=x\): for example, \(2^{-2}=\frac14\), \(2^1=2\), and \(2^2=4\). Each pair therefore satisfies \(y=\log_2(x)\). The graph rises from very negative values near \(x=0\), passes through \((1,0)\), and continues upward as \(x\) increases.
Worked Example: Graph \(y=\log_{\frac13}(x)\)
The base \(b=\frac13\) satisfies \(0<b<1\), so this graph is decreasing. As with every logarithmic graph, its domain is \((0,\infty)\), its range is \(\mathbb{R}\), its \(x\)-intercept is \((1,0)\), it has no \(y\)-intercept, and its vertical asymptote is \(x=0\).
Choose \(y=-2,-1,0,1,2\). Since \(x=(\frac13)^y\), the corresponding inputs are:
Thus the points are \((9,-2)\), \((3,-1)\), \((1,0)\), \((\frac13,1)\), and \((\frac19,2)\). Check, for example, that \((\frac13)^2=\frac19\) and \((\frac13)^{-1}=3\); these verify the points \((\frac19,2)\) and \((3,-1)\). The points also show the decreasing behavior: as \(x\) moves from \(\frac19\) to \(9\), \(y\) moves from \(2\) down to \(-2\). Near \(x=0\), the output grows upward rather than downward.
Worked Example: Identify Features from Exact Points
Suppose a logarithmic graph contains the points \((4,1)\), \((1,0)\), and \((\frac14,-1)\), and is of the form \(y=\log_b(x)\). Use the points to identify its base and describe the graph.
At the point \((4,1)\), \(y=1\), so the defining equation \(b^y=x\) gives \(b^1=4\). Therefore \(b=4\). Check with the point \((\frac14,-1)\): \(4^{-1}=\frac14\), as required. The point \((1,0)\) also checks because \(4^0=1\).
The function is therefore \(y=\log_4(x)\). Since \(4>1\), it is increasing. Its domain is \((0,\infty)\), its range is \(\mathbb{R}\), its \(x\)-intercept is \((1,0)\), and it has no \(y\)-intercept. Its vertical asymptote is \(x=0\). The three given points agree with each feature: all have positive \(x\)-coordinates, the intercept occurs at \((1,0)\), and the \(y\)-values increase as the displayed \(x\)-values increase from \(\frac14\) to \(4\).
Comparing the Two Possible Shapes
The base separates logarithmic graphs into two cases. The domain, range, intercept, and asymptote do not change between the cases; only the direction of the graph changes. The table summarizes the features that are most useful when sketching.
| Feature | \(b>1\) | \(0<b<1\) |
|---|---|---|
| Domain | \((0,\infty)\) | \((0,\infty)\) |
| Range | \(\mathbb{R}\) | \(\mathbb{R}\) |
| \(x\)-intercept | \((1,0)\) | \((1,0)\) |
| Vertical asymptote | \(x=0\) | \(x=0\) |
| Behavior near \(x=0\) | Falls without bound | Rises without bound |
| Behavior as \(x\) increases | Increases | Decreases |
A Common Mistake
A frequent error is to assume that every logarithmic graph increases because logarithms are being thought of as inverses of increasing exponential functions. That assumption overlooks bases between \(0\) and \(1\).
For example, it would be wrong to claim that \(y=\log_{\frac13}(x)\) increases because the input \(x\) increases. Check two inputs instead:
because \((\frac13)^0=1\) and \((\frac13)^{-1}=3\). As the input increases from \(1\) to \(3\), the output falls from \(0\) to \(-1\). The graph is decreasing, not increasing. The correction is to check the base first: \(b>1\) means increasing, while \(0<b<1\) means decreasing.
Sketching Checklist
For a reliable sketch, determine the base case before placing points. Then mark the fixed features and use exact powers to check the graph's shape.
Confirm \(b>0\) and \(b\ne1\); decide whether \(b>1\) or \(0<b<1\).
Draw the graph only for \(x>0\), with vertical asymptote \(x=0\).
Plot \((1,0)\), then make the graph increasing for \(b>1\) or decreasing for \(0<b<1\).
Choose convenient exponents \(y\), calculate \(x=b^y\), and verify each point with \(b^y=x\).
The graph's range is all real numbers, so it continues indefinitely upward and downward overall, even though its \(x\)-values remain positive. A careful sketch should show that it approaches the \(y\)-axis without crossing it.
Check Your Understanding
Answer these questions using the base and the defining relationship \(b^y=x\).
- For \(y=\log_5(x)\), state the domain, range, intercept, vertical asymptote, and direction of change.
- Does \(y=\log_{\frac14}(x)\) increase or decrease? What happens to its output as positive \(x\) approaches zero?
- Why does no logarithmic graph \(y=\log_b(x)\) have a \(y\)-intercept?
- Find two exact points on \(y=\log_3(x)\) other than \((1,0)\), and check each using the exponential equation.
- A graph \(y=\log_b(x)\) passes through \((8,3)\). Find \(b\), and state whether the graph increases or decreases.
