Introduction to Logarithms

In this lesson, students, you will learn one of the most useful ideas in algebra: logarithms 🔍. Logarithms help us answer questions like, “What power of $10$ gives $1000$?” or “How long does it take for money to grow if it increases by a fixed percentage?” They are the reverse process of exponentiation, and they appear in science, finance, computing, and data measurement.

Learning goals

By the end of this lesson, students, you should be able to:

explain what a logarithm means and use the correct terminology,
convert between exponential form and logarithmic form,
evaluate simple logarithms,
use the laws of logarithms in algebraic manipulation,
understand how logarithms connect to exponential growth and the wider Number and Algebra topic.

Logarithms are a core part of IB Mathematics Analysis and Approaches SL because they combine symbolic manipulation with reasoning about numbers, patterns, and functions. They also help us solve equations where the unknown is in an exponent.

What is a logarithm?

A logarithm tells us the exponent needed to get a certain number from a base. If $b^x=a$, then the logarithm tells us that $x$ is the power we raise $b$ to in order to get $a$.

This is written as:

$$\log_b(a)=x \quad \text{if and only if} \quad b^x=a$$

Here:

$b$ is the base,
$a$ is the argument or number inside the logarithm,
$x$ is the logarithm itself.

For example, since $2^3=8$, we can write:

$$\log_2(8)=3$$

This means: “What power of $2$ gives $8$?” The answer is $3$.

Another example is:

$$\log_{10}(1000)=3$$

because $10^3=1000$.

This idea is especially useful when the exponent is unknown. For example, if $2^x=32$, we can rewrite $32$ as $2^5$, so $x=5$. In logarithmic form, that same fact becomes:

$$x=\log_2(32)$$

The connection between exponential and logarithmic form

Logarithms and exponents are inverse operations. That means they “undo” each other, just like addition and subtraction, or multiplication and division.

$$b^x=a,$$

then

$$\log_b(a)=x.$$

This relationship is central to logarithms. It allows you to switch between two forms of the same equation.

Example 1: converting from exponential to logarithmic form

$$5^2=25,$$

then the logarithmic form is

$$\log_5(25)=2.$$

Example 2: converting from logarithmic to exponential form

$$\log_3(81)=4,$$

then the exponential form is

$$3^4=81.$$

These conversions are not just a memorization task, students. They help you recognize structure in equations, which is a major skill in algebra.

Common logarithms and natural logarithms

In mathematics, some bases are used so often that they have special names.

Common logarithm

The common logarithm uses base $10$ and is written as:

$$\log(x)$$

or sometimes

$$\log_{10}(x).$$

For example,

$$\log(100)=2$$

because $10^2=100$.

Natural logarithm

The natural logarithm uses base $e$, where $e$ is an irrational number approximately equal to $2.71828$. It is written as:

$$\ln(x)$$

and means

$$\log_e(x).$$

For example,

$$\ln(e^5)=5.$$

Natural logarithms are especially important in calculus, continuous growth models, and many applications in science.

Important conditions for logarithms

Not every number can go inside a logarithm. For a real logarithm:

$$\log_b(a)$$

is only defined when:

$b>0$,
$b\neq 1$,
$a>0$.

These conditions matter because exponentiation with a positive base behaves in a certain way, and logarithms reverse that behavior.

Why is $a>0$ required?

If $b>0$, then $b^x$ is always positive for real $x$. So the logarithm can only “undo” positive outputs. That means numbers like $0$ or $-3$ cannot be written as real logarithms.

For example, $\log_2(-8)$ is not a real number.

Why can’t the base be $1$?

Because $1^x=1$ for every real $x$. That would make the inverse impossible: many exponents would give the same result, so the logarithm would not be a function.

Evaluating logarithms

To evaluate a logarithm, students, ask: “What exponent gives this number?”

Example 3

$$\log_4(64)=x$$

We want the power of $4$ that gives $64$. Since

$$4^3=64,$$

we get

$$\log_4(64)=3.$$

Example 4

$$\log_{\frac12}(8)=x$$

This means:

$$\left(\frac12\right)^x=8.$$

Since $8=2^3$ and $\frac12=2^{-1}$, we can write:

$$\left(2^{-1}\right)^x=2^3$$

$$2^{-x}=2^3$$

and therefore

$$x=-3.$$

So,

$$\log_{\frac12}(8)=-3.$$

This example shows that logarithms can be negative, especially when the base is between $0$ and $1$.

Laws of logarithms

Logarithms have rules that make algebra easier. These laws are very important in IB Mathematics Analysis and Approaches SL.

Product law

$$\log_b(MN)=\log_b(M)+\log_b(N)$$

This works because multiplying inside a logarithm corresponds to adding exponents.

Quotient law

$$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$$

This works because dividing inside a logarithm corresponds to subtracting exponents.

Power law

$$\log_b(M^k)=k\log_b(M)$$

This works because raising a power to another power multiplies exponents.

Example 5

Simplify

$$\log_2(8x)$$

Using the product law:

$$\log_2(8x)=\log_2(8)+\log_2(x).$$

Since $\log_2(8)=3$, this becomes

$$\log_2(8x)=3+\log_2(x).$$

Example 6

Simplify

$$\log_3(27y^2)$$

First split the product:

$$\log_3(27y^2)=\log_3(27)+\log_3(y^2).$$

Then apply the power law:

$$\log_3(y^2)=2\log_3(y).$$

Since $\log_3(27)=3$, we get

$$\log_3(27y^2)=3+2\log_3(y).$$

These laws are extremely useful for rewriting expressions, solving equations, and showing algebraic structure.

Solving simple logarithmic equations

A key reason logarithms are important is that they help solve equations where the variable is in the exponent.

Example 7

Solve

$$2^x=20.$$

The exponent is unknown, so we use logarithms:

$$x=\log_2(20).$$

This is an exact answer. If you want a decimal approximation, you can use a calculator:

$$x\approx 4.322.$$

Example 8

Solve

$$\log_5(x)=2.$$

Convert to exponential form:

$$5^2=x.$$

$$x=25.$$

When solving logarithmic equations, students, always check that the solution makes the argument positive. For example, if an equation leads to $x=-4$ inside $\log(x)$, that solution must be rejected because the logarithm would not be defined.

Real-world uses of logarithms 📈

Logarithms are not just abstract symbols. They help measure and model real situations.

Earthquakes

The Richter scale and other earthquake scales use logarithmic ideas. A small increase in the scale can represent a much larger increase in wave amplitude.

Sound intensity

Decibels use logarithms to compare sound levels. This is useful because human hearing responds to ratios, not just raw amounts.

Growth and decay

Population growth, bacterial growth, and compound interest often lead to exponential equations. Logarithms help find the time variable when the exponent is unknown.

For example, compound interest can be modeled by

$$A=P\left(1+r\right)^t,$$

and if you want to solve for $t$, logarithms are often the best tool.

Computing and data

Algorithms and data analysis often involve logarithmic patterns. For instance, some search and sorting methods grow much more slowly than linear methods, and logarithms help describe that efficiency.

Conclusion

Logarithms are a way to ask, “What exponent gives this number?” They reverse exponentiation, connect directly to exponential equations, and provide powerful tools for algebraic manipulation. In IB Mathematics Analysis and Approaches SL, students, you will use logarithms to simplify expressions, solve equations, and model real-world growth and measurement. Mastering them helps build strong skills in Number and Algebra and prepares you for more advanced mathematics.

Study Notes

A logarithm answers the question: “What power of $b$ gives $a$?”
The definition is $\log_b(a)=x$ if and only if $b^x=a$.
Converting between forms is essential:
$b^x=a \leftrightarrow \log_b(a)=x$
Common logarithm means base $10$: $\log(x)$.
Natural logarithm means base $e$: $\ln(x)=\log_e(x)$.
A real logarithm requires $b>0$, $b\neq 1$, and $a>0$.
The laws of logarithms are:
$\log_b(MN)=\log_b(M)+\log_b(N)$
$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$
$\log_b(M^k)=k\log_b(M)$
Logarithms are useful for solving equations like $2^x=20$ and $\log_5(x)=2$.
Logarithms appear in science, finance, sound, earthquakes, and computing.
In IB Mathematics Analysis and Approaches SL, logarithms support reasoning, symbolic manipulation, and modeling in Number and Algebra.