Unit 4 · Topic 2.11
Logarithmic Functions
Describe transformations, domains, ranges, intercepts, and asymptotes of logarithmic functions. Develop the idea through symbolic, numerical, graphical, and contextual representations.
Learning Goals
- Analyze transformed logarithmic functions.
- Determine domain, asymptote, intercepts, direction, and concavity.
- Relate logarithmic features to inverse exponentials.
1. Essential Structure
Describe transformations, domains, ranges, intercepts, and asymptotes of logarithmic functions.
Read the formula together with its domain, units, starting input, and the interval length over which change is measured.
2. Core Ideas
- For \(a\log_b(x-h)+k\), the argument requires \(x>h\) and the asymptote is \(x=h\).
- The range is all real numbers.
- Direction depends on both \(a\) and the base.
- A logarithmic graph has no turning point or inflection point.
3. Graph and Representation
Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.
4. Original Worked Example
Describe the domain and vertical asymptote of \(f(x)=\log_3(x-2)\).
Check the result in the original representation and explain its meaning, including units when the quantities are contextual.
5. AP Reasoning Workflow
- Solve the argument inequality.
- Locate the vertical asymptote.
- Determine direction and concavity.
- Find intercepts.
- Describe end behavior from within the domain.
A strong AP response shows the mathematical evidence first and then states a precise conclusion.
6. Extended Example and Application
For \(-2\log_3(x+4)+1\), the domain is \(x>-4\), the asymptote is \(x=-4\), and the graph decreases. Its x-intercept is \(x=\sqrt3-4\).
7. Technology and Validation
Graphing software may not draw the asymptote, so state it from the argument boundary.
When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.
8. Know the Parent Logarithmic Graph
The parent function \(f(x)=\log_bx\), where \(b>0\) and \(b\ne1\), has domain \((0,\infty)\), range \(( -\infty,\infty)\), vertical asymptote \(x=0\), and x-intercept \((1,0)\).
| Base | Direction | Concavity | Key points |
|---|---|---|---|
| \(b>1\) | Increasing | Concave down | \((1/b,-1),(1,0),(b,1)\) |
| \(0<b<1\) | Decreasing | Concave up | \((1/b,-1),(1,0),(b,1)\) |
The same point formulas work in both cases, but their left-to-right order changes because \(b<1\) places \(b\) to the left of 1. Logarithmic functions are continuous on their domains and have neither a turning point nor a point of inflection.
9. Track Direction and Concavity after Vertical Scaling
For \(f(x)=a\log_bx\), multiplying by a negative \(a\) reflects the graph across the x-axis. This reverses both direction and concavity.
| Base and coefficient | Direction | Concavity |
|---|---|---|
| \(b>1,\ a>0\) | Increasing | Down |
| \(b>1,\ a<0\) | Decreasing | Up |
| \(0<b<1,\ a>0\) | Decreasing | Up |
| \(0<b<1,\ a<0\) | Increasing | Down |
Horizontal and vertical translations move the graph but do not change whether it is increasing, decreasing, concave up, or concave down. A graph can be increasing and concave down at the same time: it rises while its average rates of change become smaller.
10. Find the Domain and Vertical Asymptote
For a transformed logarithm
the argument must be positive:
The argument equals zero at \(x=h\), so \(x=h\) is the vertical asymptote. The sign of \(c\) determines which side belongs to the domain.
| Function | Domain condition | Domain | Asymptote |
|---|---|---|---|
| \(\log_2(x-4)\) | \(x-4>0\) | \((4,\infty)\) | \(x=4\) |
| \(3\log_5(2-x)-1\) | \(2-x>0\) | \(( -\infty,2)\) | \(x=2\) |
| \(-\log_3(4x+6)\) | \(4x+6>0\) | \(( -3/2,\infty)\) | \(x=-3/2\) |
Neither the outside coefficient \(a\) nor the vertical shift \(k\) changes the argument restriction.
11. Locate Key Points and Intercepts
Choose argument values \(1/b\), 1, and \(b\) to generate convenient points. For
the argument is \(x-1\). The resulting points are
| Argument \(x-1\) | x-value | Log value | Point on \(f\) |
|---|---|---|---|
| \(1/3\) | \(4/3\) | -1 | \((4/3,-6)\) |
| 1 | 2 | 0 | \((2,-4)\) |
| 3 | 4 | 1 | \((4,-2)\) |
For the x-intercept, solve \(2\log_3(x-1)-4=0\):
There is no y-intercept because \(x=0\) is outside the domain \(x>1\). Always check intercept candidates against the logarithmic domain.
12. Describe End Behavior without Crossing the Asymptote
A logarithmic function is unbounded at both ends of its domain, but which end approaches \(\infty\) depends on direction and on the side of the vertical asymptote.
| Parent function | Near \(x=0\) from the right | As \(x\to\infty\) |
|---|---|---|
| \(\log_bx,\ b>1\) | \(f(x)\to-\infty\) | \(f(x)\to\infty\) |
| \(\log_bx,\ 0<b<1\) | \(f(x)\to\infty\) | \(f(x)\to-\infty\) |
For a transformed graph, first identify the domain side and direction. A vertical shift changes neither unbounded end behavior nor the asymptote location. Since the range is all real numbers, a general logarithmic graph has no horizontal asymptote.
A vertical asymptote is a boundary approached from within the domain; it is not automatically approached from both sides.
13. Recognize Proportional Inputs over Equal Output Steps
For \(f(x)=\log_bx\), equal changes in output correspond to multiplicative changes in input. If the output rises by 1, the input is multiplied by \(b\).
| Output \(y\) | Input for \(y=\log_2x\) | Input ratio |
|---|---|---|
| 0 | 1 | -- |
| 1 | 2 | 2 |
| 2 | 4 | 2 |
| 3 | 8 | 2 |
For the shifted function \(g(x)=\log_2(x-5)\), outputs \(0,1,2,3\) occur at raw inputs \(6,7,9,13\). Those raw inputs are not proportional, but their adjusted distances from the asymptote,
are proportional. This pattern helps identify an additive transformation of a logarithmic function from a table.
14. Complete Analysis of a Transformed Logarithm
Analyze
- Domain: \(2-x>0\), so \(x<2\).
- Asymptote: the argument approaches zero at \(x=2\).
- Range: all real numbers.
- Direction: the inside reflection makes \(\log_3(2-x)\) decrease; multiplying by -2 makes \(f\) increase.
- Concavity: the inside reflection preserves concavity down, while the negative outside factor changes it to concave up.
- x-intercept: \(-2\log_3(2-x)+1=0\) gives \(x=2-\sqrt3\).
- y-intercept: \(f(0)=1-2\log_3(2)\approx-0.262\).
The end behavior is
AP-ready description: state the domain and asymptote first, then justify direction and concavity using all transformations rather than relying only on the sign of the outside coefficient.
15. Common Errors
- Do not shift the asymptote in the opposite direction from \(x-h\).
- Reversing the domain shift in \(\log_b(x-h)\).
- Ignoring a negative outside coefficient when deciding direction.
- Giving a numerical result without a domain check, units, or interpretation.
Key Takeaways
- Describe transformations, domains, ranges, intercepts, and asymptotes of logarithmic functions.
- Core relationship: \(f(x)=a\log_b(x-h)+k\)
- Error check: Do not shift the asymptote in the opposite direction from \(x-h\).
- Analyze \(3\ln(2-x)-5\).
- Find the x-intercept of \(\log_2(x-1)-3\).
- Describe the effect of multiplying a basic logarithm input by its base.