# Introduction to logarithms (base 10 and e)

> IB Mathematics: Applications and Interpretation SL · Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-simplification-of-logarithmic-expressions/

This sub-topic introduces logarithms as the inverse of exponentials. You will learn the relationship $a^x = b \Leftrightarrow \log_a b = x$, recognize common (base 10) and natural (base $e$) logarithms, and evaluate logarithms numerically using your GDC.

**Prerequisites:** [Laws of exponents with integer exponents](https://www.owlsprep.com/study/ib-math-ai-sl-u1-laws-of-exponents-with-rational/)

## Learning objectives

- Understand a logarithm as the inverse of an exponential
- Use the relationship $a^x = b \Leftrightarrow \log_a b = x$ to convert between exponential and logarithmic form
- Recognize common logarithms (base 10) and natural logarithms (base $e$)
- Evaluate logarithms numerically using a GDC

## What Is a Logarithm?

A logarithm answers the question: to what power must we raise a base to get a given number? If $a^x = b$, then $x$ is the logarithm of $b$ to base $a$. So a logarithm is simply the inverse operation of raising a base to a power.

**Definition of a Logarithm** — For a base $a>0$, $a \neq 1$, and argument $b>0$: $\log_a b = x$ means exactly that $a^x = b$. The base of the log matches the base of the power, and the log gives the exponent.

*Notation:* \log_a b = x \iff a^x = b

*Example:* $\log_2 8 = 3$ because $2^3 = 8$

This equivalence lets you rewrite any exponential statement as a logarithm and vice versa. Reading the relationship both ways is the single most useful skill for this topic.

**Worked example:** Rewrite $10^3 = 1000$ in logarithmic form, and rewrite $\log_5 25 = 2$ in exponential form.

1. For $10^3 = 1000$, the base is 10, the exponent is 3, and the result is 1000. Using $a^x = b \Leftrightarrow \log_a b = x$:
2. $$10^3 = 1000 \iff \log_{10} 1000 = 3$$
3. For $\log_5 25 = 2$, the base is 5, the log value (exponent) is 2, and the argument is 25:
4. $$\log_5 25 = 2 \iff 5^2 = 25$$

> **warning**
>
> The argument of a logarithm must be positive. $\log_a b$ is only defined for $b>0$, because a positive base raised to any real power can never be zero or negative.

> **Exam tip:** If you are stuck on a logarithm, immediately rewrite it as a power: $\log_a b = x$ becomes $a^x = b$. This turns an unfamiliar log into a familiar exponent question.

## Common and Natural Logarithms

Two bases are used so often that they have their own shorthand. Base 10 gives the common logarithm, written $\log b$ (sometimes $\lg b$). Base $e$ (where $e \approx 2.718$) gives the natural logarithm, written $\ln b$.

**Common and Natural Logarithms** — Common logarithm: $\log b = \log_{10} b$, so $\log b = x$ means $10^x = b$. \\ Natural logarithm: $\ln b = \log_e b$, so $\ln b = x$ means $e^x = b$.

*Example:* $\log 100 = 2$ because $10^2 = 100$; $\ln e = 1$ because $e^1 = e$

Natural logarithms appear throughout IB AI SL when modelling exponential growth and decay, because continuous growth is naturally described using base $e$. Common logarithms appear in applied scales such as pH, sound intensity, and the Richter scale.

## Evaluating Logarithms with Technology

Some logarithms give exact whole-number values that you can find by inspection, such as $\log 1000 = 3$. Most logarithms, however, are not whole numbers and are evaluated numerically using your GDC.

1. For a base-10 logarithm, use the $\log$ key: e.g. $\log 250 \approx 2.40$.
2. For a natural logarithm, use the $\ln$ key: e.g. $\ln 12 \approx 2.48$.
3. Most GDCs also have a general $\log_a b$ template that lets you enter any base directly.

**Worked example:** Use a GDC to evaluate $\log 45$ and $\ln 20$, each to 3 significant figures.

1. Enter $\log 45$ using the log key (base 10):
2. $$\log 45 = 1.6532\ldots \approx 1.65$$
3. Enter $\ln 20$ using the ln key (base $e$):
4. $$\ln 20 = 2.9957\ldots \approx 3.00$$
5. Check each answer makes sense: $10^{1.65} \approx 45$ and $e^{3.00} \approx 20$, both consistent.

**Check your understanding**

Test your understanding:

1. What is the value of $\log_4 16$?

   - 2
   - 4
   - 8
   - 64

   *Why:* Correct! $\log_4 16 = 2$ because $4^2 = 16$. Rewriting the log as a power makes this quick.

## Common pitfalls

- **Wrong:** Reading $\log_a b$ as $a^b$
  - Why it fails: A logarithm returns the exponent, not the power. $\log_a b$ is the number $x$ such that $a^x = b$.
  - Correct: Use $\log_a b = x \Leftrightarrow a^x = b$: for example $\log_2 8 = 3$, not $2^8$.
- **Wrong:** Swapping the base and the argument, e.g. writing $\log_8 2 = 3$
  - Why it fails: The base sits at the bottom and the argument is inside. $\log_2 8 = 3$, but $\log_8 2 = \frac{1}{3}$.
  - Correct: Keep the base as the number being raised to a power: $2^3 = 8$ gives $\log_2 8 = 3$.
- **Wrong:** Taking the logarithm of a negative number or zero, e.g. $\log(-5)$
  - Why it fails: A positive base raised to any real power is always positive, so a log of a non-positive number does not exist.
  - Correct: State that $\log b$ is only defined for $b>0$.
- **Wrong:** Assuming the $\log$ key on a GDC means natural log
  - Why it fails: On most calculators the $\log$ key is base 10 and the separate $\ln$ key is base $e$; mixing them gives wrong values.
  - Correct: Use $\log$ for base 10 and $\ln$ for base $e$, or the general $\log_a b$ template for any base.

## Cheatsheet

| Idea | Statement | Notes |
| --- | --- | --- |
| Definition | $\log_a b = x \iff a^x = b$ | $a>0, a \neq 1, b>0$ |
| Common log | $\log b = \log_{10} b$ | Use the $\log$ key |
| Natural log | $\ln b = \log_e b$ | Use the $\ln$ key, $e \approx 2.718$ |
| Log of the base | $\log_a a = 1$ | Since $a^1 = a$ |
| Log of 1 | $\log_a 1 = 0$ | Since $a^0 = 1$ |
| Valid argument | $b > 0$ | Log undefined for $b \leq 0$ |

## What's next

Understanding logarithms as the inverse of exponentials is the foundation for solving exponential equations, where you take a logarithm of both sides to bring an unknown down from the exponent. This skill is heavily tested across both IB AI SL papers and appears in applied contexts such as exponential growth and decay, compound interest, and half-life problems.

- [Solving exponential equations using logarithms](https://www.owlsprep.com/study/ib-math-ai-sl-u1-solving-exponential-equations-using-logarithms/)
- [Approximation, absolute and percentage error](https://www.owlsprep.com/study/ib-math-ai-sl-u1-approximation-absolute-and-percentage-error/)

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