# Exponential and logarithmic functions

> IB Mathematics: Analysis and Approaches HL · IB AA HL
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-exponential-and-logarithmic-functions/

This module covers core properties, graphs, algebraic laws, and problem-solving for exponential and logarithmic functions, foundational for calculus and modeling applications in IB AA HL.

**Prerequisites:** [Indices and exponent rules](https://www.owlsprep.com/study/ib-math-aa-hl-u2-indices-and-exponents/); [Inverse functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-inverse-functions/)

## Learning objectives

- Identify key properties of exponential and logarithmic functions including domain, range, and asymptotes
- Apply logarithm laws to simplify expressions and solve equations
- Sketch graphs of exponential and logarithmic functions and their transformations
- Solve exponential and logarithmic equations, identifying extraneous solutions

## Definition and Core Properties of Exponential Functions

**Exponential Function** — A function with a constant positive base and variable exponent, defined for all real input values.

*Notation:* f(x) = a^x, \quad a > 0, a \neq 1

*Example:* $2^x$, $e^x$, $(\frac{1}{3})^x$

All exponential functions share key properties: they are continuous for all real $x$, they pass through the point $(0, 1)$ (since $a^0 = 1$ for any $a \neq 0$), and have a horizontal asymptote at $y=0$ for untransformed functions. If $a>1$ the function is strictly increasing, if $0<a<1$ it is strictly decreasing.

**Worked example:** For $f(x) = 3^{(x-2)} + 1$, state the domain, range, and equation of the horizontal asymptote.

1. Step 1: The exponent $x-2$ can take any real value, so horizontal shifts do not change the domain of an exponential function.
2. Domain of $f(x)$ is all real numbers: $(-\infty, \infty)$
3. Step 2: The range of the base exponential $3^{u}$ (for any real $u$) is $(0, \infty)$.
4. Step 3: The function is shifted up 1 unit, so the lower bound of the range becomes $0 + 1 = 1$, with no upper bound.
5. Range is $(1, \infty)$
6. Step 4: As $x \to -\infty$, $3^{x-2} \to 0$, so $f(x) \to 1$, so the horizontal asymptote is $y=1$.

> **Exam tip:** Always account for vertical shifts when finding the asymptote and range of transformed exponential functions — this is a common marking point.

## Logarithms as Inverses and Logarithm Laws

**Logarithm** — The inverse of the exponential function $a^y = x$, defined only for positive $x$, where $a>0, a \neq 1$.

*Notation:* \log_a x = y

*Example:* $\log_2 8 = 3$, since $2^3 = 8$

As inverses of exponentials, logarithms swap the domain and range of exponentials: domain is $x>0$, range is all real numbers. All core logarithm laws are derived directly from exponent rules.

- Product law: $\log_a (xy) = \log_a x + \log_a y$
- Quotient law: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
- Power law: $\log_a (x^k) = k \log_a x$
- Change of base: $\log_a b = \frac{\log_c b}{\log_c a}$ for any $c>0, c \neq 1$

> **mnemonic**
>
> Log laws turn operations up a level: multiplication becomes addition, division becomes subtraction, exponents become multiplication.

**Worked example:** Simplify $2 \ln 3 + \ln 4 - \ln 12$ to a single logarithm.

1. Step 1: Apply the power law to the first term:
2. $$2 \ln 3 = \ln 3^2 = \ln 9$$
3. Step 2: Apply the product law to add the first two terms:
4. $$\ln 9 + \ln 4 = \ln (9 \times 4) = \ln 36$$
5. Step 3: Apply the quotient law to subtract the final term:
6. $$\ln 36 - \ln 12 = \ln \left(\frac{36}{12}\right) = \ln 3$$

## Graphs and Transformations

Since exponential and logarithmic functions are inverses, their graphs are reflections of each other over the line $y=x$. Key features for $a>1$ (the most common case in exams) are:

| Function | Asymptote | Key Point | Monotonicity |
| --- | --- | --- | --- |
| $y = a^x$ | Horizontal $y=0$ | $(0, 1)$ | Strictly increasing |
| $y = \log_a x$ | Vertical $x=0$ | $(1, 0)$ | Strictly increasing |

Transformations follow the same rules as all functions: horizontal shifts move asymptotes horizontally, vertical shifts move them vertically. Always find the new asymptote first before plotting points.

**Worked example:** For $y = \log_2 (x+1) - 2$, state the asymptote and intercepts.

1. Step 1: Start with base $y = \log_2 x$, which has vertical asymptote $x=0$.
2. Step 2: Replace $x$ with $x+1$, which shifts the graph left 1 unit. Asymptote becomes $x = -1$.
3. Step 3: Find the y-intercept by setting $x=0$: $y = \log_2 (0+1) - 2 = 0 - 2 = -2$, so y-intercept is $(0, -2)$.
4. Step 4: Find the x-intercept by setting $y=0$:
5. $$0 = \log_2(x+1) - 2 \implies \log_2(x+1) = 2 \implies x+1 = 2^2 = 4 \implies x=3$$
6. So x-intercept is $(3, 0)$.

> **Exam tip:** Shifted asymptotes are almost always a required marking point in graphing questions for these functions.

## Solving Exponential and Logarithmic Equations

For exponential equations where bases cannot be matched, take logarithms of both sides to bring down the exponent. For logarithmic equations, combine terms into a single logarithm using the laws, then convert to exponential form. Always check for extraneous solutions that fall outside the domain of the original function.

> **warning**
>
> Any solution that makes the argument of a logarithm negative or zero is invalid, even if it comes from correct algebraic manipulation.

**Worked example:** Solve $2e^{2x} - 5e^x + 2 = 0$ for $x$.

1. Step 1: Recognize this is a quadratic equation in $e^x$. Let $u = e^x$, so substitute:
2. $$2u^2 - 5u + 2 = 0$$
3. Step 2: Factor the quadratic:
4. $$(2u - 1)(u - 2) = 0 \implies u = \frac{1}{2} \text{ or } u = 2$$
5. Step 3: Substitute back $u = e^x$ and take natural logs of both sides:
6. $$e^x = \frac{1}{2} \implies x = \ln \left(\frac{1}{2}\right) = -\ln 2$$
7. $$e^x = 2 \implies x = \ln 2$$
8. Step 4: Check: $e^x$ is always positive, so both solutions are valid. Final solutions are $x = \pm \ln 2$.

*Calculator:* allowed for checking only

## Common pitfalls

- **Wrong:** Claiming $\log_a(x + y) = \log_a x + \log_a y$
  - Why it fails: Students confuse the product law for logarithms, which applies to products of arguments, not sums inside the logarithm.
  - Correct: Remember $\log_a(xy) = \log_a x + \log_a y$, this rule does not extend to $\log_a(x+y)$.
- **Wrong:** Forgetting to check for extraneous solutions when solving logarithmic equations
  - Why it fails: Algebraic manipulation can produce solutions that are outside the domain of the original logarithmic function.
  - Correct: Substitute all candidate solutions back into the original equation to confirm all logarithm arguments are positive.
- **Wrong:** Stating the domain of $a^x$ is $x \geq 0$
  - Why it fails: Confusing exponential functions (variable exponent) with power functions (variable base), which often have restricted domains.
  - Correct: The domain of any exponential function $a^x$ ($a>0$) is all real numbers $(-\infty, \infty)$.
- **Wrong:** Ignoring that $e^{\ln x} = x$ only holds for $x>0$
  - Why it fails: Students assume inverse relations hold for all inputs, but inverses are only defined for inputs in the domain of the inner function.
  - Correct: Any solution for $x \leq 0$ when working with $e^{\ln x}$ or $\log_a(a^x)$ is automatically invalid.

## Cheatsheet

| Feature | Exponential $a^x$, $a>1$ | Logarithm $\log_a x$, $a>1$ |
| --- | --- | --- |
| Domain | $(-\infty, \infty)$ | $x > 0$ |
| Range | $y > 0$ | $(-\infty, \infty)$ |
| Asymptote | Horizontal: $y=0$ | Vertical: $x=0$ |
| Key Point | $(0, 1)$ | $(1, 0)$ |
| Monotonicity | Strictly increasing | Strictly increasing |
| Product Rule | $a^m a^n = a^{m+n}$ | $\log_a(xy) = \log_a x + \log_a y$ |
| Quotient Rule | $a^m/a^n = a^{m-n}$ | $\log_a(x/y) = \log_a x - \log_a y$ |
| Power Rule | $(a^m)^n = a^{mn}$ | $\log_a x^k = k \log_a x$ |
| Inverse Relation | $a^{\log_a x} = x \ (x>0)$ | $\log_a(a^x) = x \ (\forall x)$ |

## What's next

Exponential and logarithmic functions are foundational for nearly all remaining topics in IB AA HL. They are critical for calculus, where you will learn to differentiate and integrate these functions, and they are the core tool for modeling real-world phenomena like population growth, radioactive decay, and compound interest. They also appear frequently in differential equations, a key topic for Paper 3. Mastering their properties and algebraic manipulation now will pay dividends for more advanced topics later in the course.

- [Basic trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-basic-trigonometric-functions/)
- [Inverse trigonometric functions (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u2-inverse-trigonometric-functions/)
- [Trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-hl-u2-trigonometric-identities/)

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