How to Graph Logarithmic Functions?
TL;DR: A logarithmic graph is the inverse of an exponential graph. The parent graph \(y=\log_b(x)\) has domain \(x>0\), range all real numbers, vertical asymptote \(x=0\), and key points \((1,0)\), \((b,1)\), and \((b^2,2)\).
- If \(b>1\), the parent log graph increases. If \(0
- For \(y=a\log_b(x-h)+k\), the vertical asymptote moves to \(x=h\).
- Use easy log inputs \(1\), \(b\), and \(b^2\) to build reliable points quickly.
- A log graph often has no y-intercept because the input of the logarithm must be positive.
Graphing logarithmic functions is much easier when you stop hunting for random x-values. A log graph has a wall, called a vertical asymptote, and it has a few dependable points that come from powers of the base. In this lesson, we will use those pieces like a tutor would: find the asymptote, build easy points, apply the transformations, and check the shape.
Graph Logarithmic Functions: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Translate firstAsk: the base to what power gives the input?
- Use rules legallyProducts, quotients, and powers have rules; sums do not split.
- Protect the domainKeep the log input positive and track asymptotes when graphing.
Worked examples
Evaluate a log
- Ask 3 to what power equals 81.
- 3 to the fourth power is 81.
- The logarithm is that exponent.
Find a log domain
- The input is x – 5.
- Require x – 5 > 0.
- Solve the inequality.
Try one before moving on
Graph Logarithmic Functions: pop-up practice
What a logarithmic graph means
A logarithm asks an exponent question:
Free tool: try the problems from this lesson in our Logarithm Calculator.
So \(\log_2(8)=3\) because \(2^3=8\). On a graph, the input \(x\) must be positive, because there is no real exponent that makes a positive base equal to zero or a negative number.
The parent graph \(y=\log_b(x)\)
| Feature | What it means |
|---|---|
| Base | \(b>0\) and \(b\ne1\). Bases greater than 1 increase; bases between 0 and 1 decrease. |
| Domain | \(x>0\), because the log input must be positive. |
| Range | All real numbers. The graph can go up or down without a highest or lowest y-value. |
| Vertical asymptote | \(x=0\). The graph approaches the y-axis but never touches it. |
| Key points | \((1,0)\), \((b,1)\), and \((b^2,2)\). These come from \(\log_b(1)=0\), \(\log_b(b)=1\), and \(\log_b(b^2)=2\). |
| Inverse relationship | \(y=\log_b(x)\) reflects \(y=b^x\) across \(y=x\). |
| One-to-one | Every horizontal line hits the graph at most once, so a logarithmic function has an inverse. |
End behavior and special log facts
The graph behavior near the asymptote depends on the base.
| Base range | As \(x\to0^+\) | As \(x\to\infty\) | Shape |
|---|---|---|---|
| \(b>1\) | \(y\to-\infty\) | \(y\to\infty\) | Increasing |
\(0| \(y\to\infty\) | \(y\to-\infty\) | Decreasing | |
Two more facts are worth keeping close:
- \(\ln(x)=\log_e(x)\), where \(e\approx2.718\).
- Change of base: \(\log_b(x)=\dfrac{\ln x}{\ln b}\). This is why graphing tools can use natural logs to graph logs with many different bases.
How to graph \(y=a\log_b(x-h)+k\)
Most log graph problems in school math are transformations of the parent graph. Use this reliable process:
- Find the vertical asymptote. Set the log input equal to zero. For \(x-h\), the asymptote is \(x=h\).
- Find the domain. Make the log input greater than zero. For \(x-h\), the domain is \(x>h\).
- Choose easy inputs. Make the inside of the log equal \(1\), \(b\), and \(b^2\).
- Transform the y-values. The parent y-values \(0\), \(1\), and \(2\) become \(k\), \(a+k\), and \(2a+k\).
- Sketch the curve. The curve should approach the asymptote and pass through your transformed points.
Those three points are the fastest clean graphing points for \(y=a\log_b(x-h)+k\).
Worked example 1: graph \(f(x)=2\log_3(x+1)\)
Step-by-step
- The input is \(x+1\), so set \(x+1=0\). The vertical asymptote is \(x=-1\).
- The domain is \(x+1>0\), so \(x>-1\).
- Use inside values \(1\), \(3\), and \(9\). That gives x-values \(0\), \(2\), and \(8\).
- The parent outputs \(0\), \(1\), and \(2\) are multiplied by 2, so the y-values are \(0\), \(2\), and \(4\).
Graphing data
Asymptote: \(x=-1\)
Domain: \((-1,\infty)\)
Range: all real numbers
Points: \((0,0)\), \((2,2)\), \((8,4)\)
The graph is increasing because the base is \(3>1\) and the coefficient is positive.
Worked example 2: graph \(g(x)=-\log_2(x-4)+1\)
This one has a horizontal shift, a reflection across a horizontal line, and a vertical shift.
- Set \(x-4=0\). The vertical asymptote is \(x=4\), and the domain is \(x>4\).
- Use inside values \(1\), \(2\), and \(4\), giving x-values \(5\), \(6\), and \(8\).
- The parent outputs are \(0\), \(1\), and \(2\).
- Multiply by \(-1\) and add \(1\), giving y-values \(1\), \(0\), and \(-1\).
Graph \((5,1)\), \((6,0)\), and \((8,-1)\), then sketch a decreasing curve approaching \(x=4\).
Worked example 3: graph \(h(x)=\log_{1/2}(x)+2\)
The base is between 0 and 1, so the graph decreases as x increases.
| Inside value | Log value | After adding 2 | Point |
|---|---|---|---|
| \(1\) | \(0\) | \(2\) | \((1,2)\) |
| \(1/2\) | \(1\) | \(3\) | \((1/2,3)\) |
| \(2\) | \(-1\) | \(1\) | \((2,1)\) |
Common mistakes to avoid
- Using random x-values first. Pick inside values \(1\), \(b\), and \(b^2\) instead. They make the log values easy.
- Forgetting the domain. The inside of the log must be greater than zero, not greater than or equal to zero.
- Drawing a y-intercept automatically. Many log graphs never touch the y-axis.
- Mixing up shifts. In \(\log_b(x-h)\), the asymptote is \(x=h\), not \(x=-h\).
- Ignoring a negative coefficient. A negative value of \(a\) flips the graph vertically.
Practice problems
- For \(f(x)=\log_4(x-2)\), find the vertical asymptote and domain.
- For \(g(x)=3\log_2(x)+1\), list three easy points.
- For \(h(x)=-2\log_5(x+3)\), find the asymptote and say whether the graph increases or decreases.
- For \(p(x)=\log_{1/3}(x)-4\), find the point that comes from the parent point \((1,0)\).
- For \(q(x)=2\log_3(x-1)-5\), find the asymptote, domain, range, three graphing points, and whether the graph increases or decreases.
Answers:
- Asymptote \(x=2\); domain \(x>2\).
- Use inside values \(1,2,4\): points \((1,1)\), \((2,4)\), \((4,7)\).
- Asymptote \(x=-3\). The graph decreases because the base is greater than 1 and the coefficient is negative.
- The point \((1,0)\) shifts down 4, so \((1,-4)\).
- Asymptote \(x=1\); domain \(x>1\); range all real numbers. Use inside values \(1,3,9\), so x-values are \(2,4,10\). Parent outputs \(0,1,2\) become \(-5,-3,-1\). Points: \((2,-5)\), \((4,-3)\), \((10,-1)\). The graph increases because \(3>1\) and the coefficient is positive.
Related EffortlessMath lessons
If a piece of this lesson feels rusty, these pages connect directly to the skills used here:
Recommended EffortlessMath Books
For a workbook that builds logarithmic graphs into a full exponential-and-log unit, the Algebra II for Beginners walks through every transformation with worked examples. For pre-calc-level coverage that prepares you for calculus and natural logs, see the Pre-Calculus for Beginners.
Frequently Asked Questions
How do you graph logarithmic functions step by step?
Start by finding the vertical asymptote: set the log input equal to zero. Next, pick easy inputs that make the log equal 0, 1 and 2 (input 1, the base b, and b²), then apply any stretch, reflection or shift to those points. Plot them and draw a smooth curve that hugs the asymptote on one side and keeps rising or falling slowly on the other.
How do you graph logarithmic functions with transformations?
Write the function as y = a·log_b(x − h) + k. The asymptote moves to x = h, k shifts every point up or down, and a stretches the graph (a negative a flips it over the x-axis). For f(x) = 2·log₃(x + 1), the asymptote is x = −1 and the parent points (1, 0) and (3, 1) become (0, 0) and (2, 2).
How do you graph logarithmic functions using a table of values?
Choose x-values that are powers of the base so the outputs come out as whole numbers. For y = log₂(x), use x = 1/2, 1, 2, 4 and 8, which give y = −1, 0, 1, 2 and 3. Plot those pairs, remember that x must stay greater than 0, and connect them with a smooth curve that drops toward the y-axis.
Is a logarithmic function increasing or decreasing?
It depends on the base. When the base is greater than 1, such as log₂(x) or log₁₀(x), the graph always increases. When the base is between 0 and 1, such as log_{1/2}(x), the graph always decreases. A negative coefficient in front of the log also flips the direction, so −log₂(x) decreases.
Are log graphs functions?
Yes. A logarithmic graph passes the vertical line test, because each allowed x-value gives exactly one y-value. It also passes the horizontal line test, which makes it one-to-one. That is why y = log_b(x) has an inverse, the exponential function y = bˣ, and the two graphs are mirror images across the line y = x.
How do you graph logarithmic functions on Desmos?
Type the function directly, for example y = log_3(x + 1). Desmos reads log_3 as base 3, log as base 10 and ln as base e. Once it is drawn, click the curve to see key points such as the x-intercept, and compare it with your hand sketch to check the asymptote and the direction of the graph.
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