Introduction to Logarithms

Welcome, students 👋 In this lesson, you will learn what logarithms are, why they exist, and how they connect to the larger world of Number and Algebra. Logarithms are an important part of IB Mathematics: Analysis and Approaches HL because they help us work with very large numbers, very small numbers, and equations where the unknown appears in an exponent. By the end of this lesson, you should be able to explain the basic ideas behind logarithms, use their notation correctly, and solve simple logarithmic and exponential problems with confidence.

Why logarithms matter

Imagine a city grows by $10\%$ each year. If the population starts at $1000$, then after one year it becomes $1000(1.1)$, after two years it becomes $1000(1.1)^2$, and so on. If you want to know how many years it will take to reach $2000$, you cannot solve that easily using only ordinary arithmetic. You need a way to “undo” the exponent. That is exactly what a logarithm does 📈

A logarithm is the inverse operation of exponentiation. If $a^x=b$, then the logarithm tells you the exponent $x$. In words, the logarithm answers: “To what power must the base $a$ be raised to get $b$?” This idea is very useful in science, finance, computer science, and statistics. For example, the Richter scale for earthquakes and the pH scale for acidity both use logarithmic ideas because they compress huge ranges of values into manageable numbers.

The main terminology is simple but very important:

$a$ is the base
$x$ is the exponent or power
$b$ is the result
$\log_a b$ means the power to which $a$ must be raised to get $b$

So the statement $a^x=b$ is equivalent to $\log_a b=x$.

The definition of a logarithm

The logarithm is defined only when the base and the input satisfy certain rules. For a real logarithm $\log_a b$:

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

These conditions matter because exponentiation with a positive base behaves in a controlled way. If the base were $1$, then $1^x$ would always be $1$, so you could not recover a unique exponent. If $b\le 0$, then there is no real logarithm, because no positive base raised to a real power gives a non-positive result.

Here are some quick examples:

$\log_{10} 100=2$ because $10^2=100$
$\log_2 8=3$ because $2^3=8$
$\log_5 1=0$ because $5^0=1$
$\log_{3} \left(\frac{1}{9}\right)=-2$ because $3^{-2}=\frac{1}{9}$

Notice that logarithms can be negative, zero, or positive depending on the number being logged. This happens because exponents can be negative, zero, or positive too.

Common and natural logarithms

Two logarithms appear very often in mathematics:

The common logarithm: $\log x$, which usually means $\log_{10} x$
The natural logarithm: $\ln x$, which means $\log_e x$

Here, $e$ is a special irrational number approximately equal to $2.71828$. It appears in continuous growth, calculus, and many advanced applications.

You should know how to interpret these notations:

$\log x$ usually means base $10$ in school mathematics unless stated otherwise
$\ln x$ always means base $e$

Examples:

$\log 1000=3$ because $10^3=1000$
$\ln(e^4)=4$ because $e^4$ has exponent $4$
$\ln(1)=0$ because $e^0=1$

When using IB-style reasoning, always check the base carefully. Misreading $\log x$ can lead to a wrong answer even if the algebra is correct.

Logarithms as inverses of exponentials

A key idea in Number and Algebra is inverse operations. Addition and subtraction are inverses, multiplication and division are inverses, and exponentiation and logarithms are inverses too. This connection helps us solve equations.

For example, consider the equation $2^x=16$. Since $16=2^4$, the solution is $x=4$. In logarithmic form, we can write

$$x=\log_2 16$$

because $\log_2 16=4$.

This gives a general solving strategy:

Rewrite the equation to isolate the exponential expression.
Convert to logarithmic form if needed.
Use the properties of logs or a calculator when appropriate.

Another example is $5^x=12$. Since $12$ is not a power of $5$, we use logarithms:

$$x=\log_5 12$$

If needed, use the change of base formula:

$$\log_a b=\frac{\log_c b}{\log_c a}$$

for any valid base $c$. In school practice, $c$ is often $10$ or $e$.

Logarithmic laws

The logarithmic laws follow from the laws of exponents. They are essential for simplifying expressions and solving equations. If $a>0$, $a\neq 1$, and $x>0$, $y>0$, then:

$$\log_a(xy)=\log_a x+\log_a y$$

$$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$$

$$\log_a(x^r)=r\log_a x$$

These laws are powerful because they turn multiplication into addition and powers into coefficients.

Example 1:

$$\log_2(8\cdot 4)=\log_2 8+\log_2 4=3+2=5$$

and indeed $8\cdot 4=32$, so $\log_2 32=5$.

Example 2:

$$\log_{10}(1000)-\log_{10}(10)=3-1=2$$

which matches

$$\log_{10}\left(\frac{1000}{10}\right)=\log_{10}(100)=2$$

Example 3:

$$\log_3(27^2)=2\log_3(27)=2\cdot 3=6$$

because $27=3^3$ and $27^2=3^6$.

Be careful: these laws only work when the inputs of the logarithms are positive. That means expressions inside logs must stay positive, which is very important when solving equations.

Solving simple logarithmic equations

A logarithmic equation is one where the unknown appears inside a logarithm. The main strategy is to rewrite the equation in exponential form and then solve.

Example 1:

$$\log_2 x=5$$

Convert to exponential form:

$$x=2^5=32$$

Example 2:

$$\ln x=3$$

Rewrite as:

$$x=e^3$$

Example 3:

$$\log_3(x-1)=2$$

Then

$$x-1=3^2=9$$

$$x=10$$

Always check the domain at the end. For instance, if an equation gives $x=-4$ but the log contains $x$, then that solution is invalid because logarithms require positive input.

Example 4:

$$\log_2(x-1)+\log_2(x+1)=3$$

Combine the logs:

$$\log_2\big((x-1)(x+1)\big)=3$$

$$x^2-1=2^3=8$$

then

$$x^2=9$$

so $x=3$ or $x=-3$. But the domain requires $x-1>0$, so $x>1$. Therefore the only valid solution is $x=3$.

This is a very IB-style habit: solve, then verify solutions against the domain ✅

Solving exponential equations using logarithms

Sometimes the unknown is in an exponent, and logarithms are the best tool to isolate it. For example:

$$3^x=20$$

Take logs of both sides:

$$\ln(3^x)=\ln(20)$$

Use the power law:

$$x\ln 3=\ln 20$$

$$x=\frac{\ln 20}{\ln 3}$$

This may not give a neat integer, but it gives an exact expression. A calculator gives an approximation.

This method is used in real life for problems like compound growth, radioactive decay, and cooling processes. If a quantity changes according to

$$A=P(1+r)^t$$

$$A=Pe^{kt}$$

logarithms let you solve for $t$. That is why logs are so useful in modeling change over time.

Connection to Number and Algebra

Logarithms fit naturally into Number and Algebra because they extend the number system and strengthen symbolic manipulation. They connect with:

exponent laws
inverse operations
solving equations
domain restrictions
algebraic simplification

They also prepare you for more advanced topics in IB Mathematics: Analysis and Approaches HL, such as functions and graphs, transformations, calculus, and mathematical modeling. For example, the graph of $y=\log_a x$ is the inverse of $y=a^x$. This means the two graphs are reflections across the line $y=x$.

Important graph features include:

the domain of $y=\log_a x$ is $x>0$
the range is all real numbers
there is a vertical asymptote at $x=0$
if $a>1$, the function is increasing
if $0<a<1$, the function is decreasing

These facts help explain why logarithms behave the way they do and why they are so closely tied to exponential growth and decay.

Conclusion

Logarithms are a powerful inverse operation that help us solve exponential equations, simplify expressions, and model real-world growth or decay. students, the most important ideas to remember are the definition $a^x=b\iff \log_a b=x$, the laws of logarithms, the meaning of $\log x$ and $\ln x$, and the need to check that logarithmic inputs are positive. In IB Mathematics: Analysis and Approaches HL, logarithms are not just a topic on their own — they are a tool that connects algebra, functions, modeling, and problem solving. Mastering them will make later topics much easier 🚀

Study Notes

A logarithm answers the question: “What exponent do I need?”
The definition is $a^x=b\iff \log_a b=x$.
For a real logarithm, $a>0$, $a\neq 1$, and $b>0$.
$\log x$ usually means $\log_{10} x$ and $\ln x$ means $\log_e x$.
The basic laws are $\log_a(xy)=\log_a x+\log_a y$, $\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$, and $\log_a(x^r)=r\log_a x$.
To solve $\log_a x=c$, rewrite it as $x=a^c$.
To solve $a^x=b$, use $x=\log_a b$ or the change of base formula.
Always check that the argument of every logarithm is positive.
Logarithms are the inverse of exponentials, so their graphs are reflections across $y=x$.
Logarithms are important for growth models, decay models, and many real-world applications.