Logarithmic Functions

Lesson ~11 min read 8 MCQs

In simple terms: In simple terms, logarithmic functions are the "undo" button for exponential functions, helping us find the exponent needed to get a certain result.

Why this matters

Have you ever heard a news report about an earthquake? They'll say something like, "A magnitude 7.1 earthquake struck off the coast of Alaska," while a smaller one might be a 3.5. But what does that number mean? A 7.1 doesn't feel twice as strong as a 3.5. In fact, it's thousands of times more powerful.

The Richter scale, which measures earthquake intensity, is logarithmic. This means that for each whole number you go up on the scale, the ground shaking increases by a factor of 10. A magnitude 4 is 10 times more powerful than a 3, and a 5 is 100 times more powerful than a 3.

Logarithms are nature's way of handling gigantic ranges of numbers, from the microscopic to the astronomical, and squishing them into a more manageable scale. Today, we'll pull back the curtain on these powerful functions and get to know their core characteristics.

Concept overview

flowchart TD
    A[Start: Analyze f(x) = a log_b(x-h) + k] --> B{Find Asymptote & Domain};
    B --> C[Set argument to zero: x - h = 0 --> Asymptote is x = h];
    C --> D[Domain is x > h or (h, ∞)];
    A --> E{Determine Shape};
    E --> F{Is b > 1?};
    F -- Yes --> G[Increasing];
    F -- No (0 < b < 1) --> H[Decreasing];
    E --> I{Check for reflection 'a'};
    I -- a > 0 --> J[Concavity depends on b];
    I -- a < 0 --> K[Concavity is flipped];
    A --> L[Range is always (-∞, ∞)];
    A --> M{End Behavior};
    M --> N[lim (x→h+) is ±∞];
    M --> O[lim (x→∞) is ±∞];

Core explanation

Alright, let's dive into the world of logarithmic functions. The single most important thing to remember is this: a logarithm is an exponent. That's it. When you see y = log₂(x), it's asking the question, "What exponent y do I need to put on the base 2 to get the number x?" This is the inverse of the exponential function x = 2^y.

Because they are inverses, their graphs are reflections of each other across the line y = x. Imagine taking the graph of an exponential function like f(x) = 2^x, which starts slow and then rockets up, and flipping it over that diagonal line. The result is the graph of g(x) = log₂(x).

Let's break down the key characteristics that emerge from this inverse relationship.

Domain, Range, and the Vertical Asymptote

Think about our exponential function, f(x) = 2^x. What kind of numbers can you get out of it? You can get any positive number, but you can never get zero or a negative number. (Try it: 2⁰ = 1, 2⁻³ = 1/8. There's no exponent you can put on 2 to get -4).

Since the logarithmic function is the inverse, we swap the inputs and outputs.

The range of f(x) = 2^x was (0, ∞). This becomes the domain of g(x) = log₂(x).
The domain of f(x) = 2^x was (-∞, ∞). This becomes the range of g(x) = log₂(x).

So, for the parent logarithmic function y = log_b(x):

Domain
(0, ∞). You can only take the log of a positive number.
Range
(-∞, ∞). The output can be any real number.

This limited domain creates a boundary. The graph of y = log₂(x) gets incredibly close to the y-axis (x=0) but never touches it. This boundary is a vertical asymptote at x = 0.

End Behavior

What happens to the graph at its edges?

As x approaches the asymptote from the right (x → 0⁺), the graph dives down to negative infinity. We write this as: lim_{x→0⁺} log₂(x) = -∞. It's trying to find the exponent you put on 2 to get a number just barely bigger than zero. That exponent has to be a huge negative number (like 2⁻¹⁰⁰⁰).
As x gets huge (x → ∞), the graph continues to rise forever. It grows much more slowly than an exponential function, but it never stops. We write this as: lim_{x→∞} log₂(x) = ∞.

Shape, Concavity, and Extrema

Look at the graph of y = 2^x. It's always increasing and always concave up. When we reflect this across y=x to get y = log₂(x), it keeps some of those properties.

The graph of y = log₂(x) is always increasing.
The graph of y = log₂(x) is always concave down.

Notice the change in concavity! The upward curve of the exponential becomes a downward curve for the logarithm. Conversely, if we started with an exponential decay function (like y = (1/2)^x), its inverse (y = log₁/₂(x)) would be always decreasing and always concave up.

Because the function is always increasing or always decreasing, it has no local maximum or minimum values (extrema). And because it's always concave up or always concave down, it has no points of inflection. The curve never changes its fundamental bend.

The Proportional Inputs Test (A Deeper Property)

This last characteristic is a bit more abstract, but it's what truly defines a function as logarithmic.

Remember how exponential functions have a constant ratio of outputs for equal-length input intervals? For y = 2^x, if you step x from 1 to 2 to 3, the outputs are 2, 4, 8. The ratio is constant: 4/2 = 2 and 8/4 = 2.

Logarithmic functions have an inverse property. For y = log₂(x), if you take equal steps in the output (say, y = 1, 2, 3), the corresponding inputs are x = 2, 4, 8. These inputs are proportional—each one is 2 times the last.

Now, here's the tricky part from the AP curriculum. What if we have a horizontal shift, like g(x) = log₂(x + 5)? Let's check if the inputs are still proportional for equal output steps.

If y=1, then 1 = log₂(x+5), so 2¹ = x+5, and x = -3.
If y=2, then 2 = log₂(x+5), so 2² = x+5, and x = -1.
If y=3, then 3 = log₂(x+5), so 2³ = x+5, and x = 3.

Are the inputs (-3, -1, 3) proportional? No. The property is broken by the additive shift.

The rule is: A function f is logarithmic if and only if its additive transformation g(x) = f(x+k) results in proportional inputs for equal-length output intervals. This is a very formal way of saying that the core "multiplying" nature is fundamental to what a logarithm is.

Worked examples

Example 1

Analyzing a Transformed Logarithmic Function

Problem: Identify the domain, range, vertical asymptote, and end behavior of the function f(x) = log₂(x - 3) + 5.

Solution:

1
Identify the parent function and transformations
The parent function is y = log₂(x). The transformations are:
- A horizontal shift 3 units to the right (x - 3).
- A vertical shift 5 units up (+ 5).
2
Find the vertical asymptote
The vertical asymptote of the parent function y = log₂(x) is x = 0. A horizontal shift moves the asymptote. Since we shifted 3 units to the right, the new asymptote is x = 3. Why? The argument of the logarithm, (x - 3), cannot be zero or negative. We find the boundary by setting it equal to zero: x - 3 = 0, which gives x = 3.
3
Determine the domain
The domain is determined by the asymptote. Since the graph exists to the right of the asymptote x = 3, the domain is (3, ∞). This is a common spot for mistakes. Students often forget that the horizontal shift changes the domain. Always set the inside of the log to be greater than zero: x - 3 > 0, so x > 3.
4
Determine the range
The range of any logarithmic function of this form is all real numbers. The vertical shift moves the graph up, but it still extends infinitely in both vertical directions. The range is (-∞, ∞).
5
Describe the end behavior
- As x approaches the asymptote from the right (x → 3⁺), the function goes to negative infinity. The +5 shift doesn't change this infinite behavior. So, lim_{x→3⁺} f(x) = -∞.
- As x goes to infinity (x → ∞), the function also goes to infinity. So, lim_{x→∞} f(x) = ∞.

Final Answer:

Vertical Asymptote: x = 3
Domain: (3, ∞)
Range: (-∞, ∞)
End Behavior: lim_{x→3⁺} f(x) = -∞ and lim_{x→∞} f(x) = ∞

Example 2

Analyzing a Reflected and Decreasing Function

Problem: Describe the key characteristics (domain, range, asymptote, concavity, and whether it's increasing or decreasing) of the function g(x) = log₁/₂(x).

Solution:

1
Analyze the base
The base is b = 1/2. Since 0 < b < 1, this will be a decreasing function. The parent exponential it's the inverse of, y = (1/2)^x, is an exponential decay function.
2
Asymptote and Domain
There are no horizontal shifts. The argument is just x. Therefore, the vertical asymptote is still x = 0, and the domain is (0, ∞).
3
Range
The range is unaffected by the base; it remains (-∞, ∞).
4
Increasing/Decreasing and Concavity
- As established, since the base is between 0 and 1, the function is always decreasing.
- Reflecting the exponential decay function y = (1/2)^x (which is always decreasing and concave up) across y=x results in a function that is always decreasing and concave up.

Final Answer: The function g(x) = log₁/₂(x) has a domain of (0, ∞), range of (-∞, ∞), and a vertical asymptote at x=0. It is always decreasing and always concave up.

Try it yourself

Ready to try a couple on your own? Don't worry about getting the perfect answer right away; focus on applying the process.

Problem 1: Consider the function g(x) = -log₄(x + 2) - 1. Identify its parent function, transformations, domain, range, and vertical asymptote. Describe its end behavior using limit notation.

Hint: Handle the transformations one by one. Where does the negative sign in front of the log take the graph? How does x+2 shift the asymptote?

Problem 2: A function f(x) is transformed into g(x) = f(x - 10). You are told that for g(x), if you increase the output y in steps of 1 (e.g., y=2, y=3, y=4), the corresponding input values x form a geometric sequence (they are proportional). What type of function is f(x)?

Hint: Refer back to the "Proportional Inputs Test." What does this special property tell you about the original function?

TL;DR

In simple terms, logarithmic functions are the "undo" button for exponential functions, helping us find the exponent needed to get a certain result.

Key terms

Logarithmic functions AP Precalculus Log function characteristics Domain and range Vertical asymptote End behavior Inverse functions Log graphs Function concavity

You can now…

2.11.A: Identify key characteristics of logarithmic functions.

Essential knowledge (exam-tested)

2.11.A.1: The domain of a logarithmic function in general form is any real number greater than zero, and its range is all real numbers.
2.11.A.2: Because logarithmic functions are inverses of exponential functions, logarithmic functions are also always increasing or always decreasing, and their graphs are either always concave up or always concave down. Consequently, logarithmic functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
2.11.A.3: The additive transformation function g(x) = f(x + k), where k ≠ 0, of a logarithmic function f in general form does not have input values that are proportional over equal-length output-value intervals. However, if the input values of the additive transformation function, g(x) = f (x + k), of any function f are proportional over equal-length output value intervals, then f is logarithmic.
2.11.A.4: With their limited domain, logarithmic functions in general form are vertically asymptotic to x = 0, with an end behavior that is unbounded. That is, for a logarithmic function in general form, lim_{x→0+} a log_b x = ±∞ and lim_{x→∞} a log_b x = ±∞.

Concept map

flowchart TD
    A[Start: Analyze f(x) = a log_b(x-h) + k] --> B{Find Asymptote & Domain};
    B --> C[Set argument to zero: x - h = 0 --> Asymptote is x = h];
    C --> D[Domain is x > h or (h, ∞)];
    A --> E{Determine Shape};
    E --> F{Is b > 1?};
    F -- Yes --> G[Increasing];
    F -- No (0 < b < 1) --> H[Decreasing];
    E --> I{Check for reflection 'a'};
    I -- a > 0 --> J[Concavity depends on b];
    I -- a < 0 --> K[Concavity is flipped];
    A --> L[Range is always (-∞, ∞)];
    A --> M{End Behavior};
    M --> N[lim (x→h+) is ±∞];
    M --> O[lim (x→∞) is ±∞];

Read what Saavi narrates

Hi everyone, it's Saavi. Let's talk about something you hear on the news all the time: earthquakes. When a reporter says a quake was a magnitude 7, and another was a 4, what's the real difference in power? It's not 3 points... it's actually 1,000 times more powerful. The scale they use is logarithmic. It's a brilliant way to take a huge range of values and make them easier to talk about.

Today, we're looking at these logarithmic functions. The key idea is simple: they are the inverses of exponential functions. They undo what exponentials do. If you understand that one relationship, all the key features, like the graph's shape and its limits, will just click into place.

Let's walk through an example together. Imagine the function f of x equals log base 2 of the quantity x minus 3, plus 5.

First, let's find the vertical asymptote. The parent function, log base 2 of x, has an asymptote at x equals 0. But our function is shifted. We look inside the parentheses... we see x minus 3. That tells us the whole graph, including the asymptote, has moved 3 units to the right. So our new vertical asymptote is at x equals 3.

This directly tells us the domain. The graph only exists to the right of that line. So the domain is from 3 to infinity, not including 3. The range is easier... for any log function like this, the range is always all real numbers, from negative infinity to positive infinity.

Now, a really common mistake I see every year is confusing the domain and the range. Students will say the domain is all real numbers. But remember, you can't take the log of a negative number or zero. That's why the domain is restricted. The *input* is restricted. The *output*, or the range, can be anything.

So, take your time, think about the inverse relationship to exponentials, and you'll find these functions are not as mysterious as they seem. You can do this.

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