Logarithmic Functions

students, imagine trying to answer this question: “What exponent gives me this number?” 🔍 That is the heart of logarithmic functions. In this lesson, you will learn how logarithms connect to exponents, why they are useful, and how to work with them in algebra, graphs, and real-world situations. By the end, you should be able to explain the meaning of a logarithm, use logarithmic laws, solve equations and inequalities involving logarithms, and connect logarithmic functions to other kinds of functions in IB Mathematics: Analysis and Approaches HL.

Introduction to logarithmic functions

A logarithm is a way of writing an exponent in reverse. If $a^x=b$, then the logarithm tells us that $x=\log_a(b)$. This means the logarithm answers the question: “To what power must the base $a$ be raised to get $b$?” 📈

For example, since $2^3=8$, we write $\log_2(8)=3$. The base is $2$, the argument is $8$, and the result is $3$. The base must satisfy $a>0$ and $a\ne 1$, and the argument must satisfy $b>0$. These conditions are important because exponential functions with positive bases always give positive outputs.

Logarithms appear naturally in many contexts. They are used to measure earthquakes with the Richter scale, sound with decibels, and pH in chemistry. In each case, logarithms help manage very large or very small numbers by turning multiplication into addition and powers into manageable expressions. That is why logarithmic functions are part of the broader study of functions: they are a powerful inverse family of exponential functions.

Understanding the notation and inverse relationship

The standard logarithmic form is $\log_a(b)=x$, which is equivalent to $a^x=b$. This equivalence is central to everything you do with logarithms. students, if you can move between these two forms confidently, you will find most logarithm problems much easier.

The common logarithm uses base $10$ and is written as $\log(x)$. The natural logarithm uses base $e$ and is written as $\ln(x)$, where $e\approx 2.71828$. In IB Mathematics, $\ln(x)$ is especially important because many models in growth and decay use base $e$.

Because logarithmic functions are inverses of exponential functions, their graphs are reflections of each other in the line $y=x$. For example, the graph of $y=2^x$ and the graph of $y=\log_2(x)$ mirror each other across $y=x$. This inverse relationship also means their domains and ranges swap:

For $y=2^x$, the domain is all real numbers and the range is $y>0$.
For $y=\log_2(x)$, the domain is $x>0$ and the range is all real numbers.

This is a key idea in function language and representation: every inverse function swaps inputs and outputs, so domain and range are exchanged.

Graphs, domain, range, and asymptotes

The graph of a logarithmic function has a characteristic shape. For $y=\log_a(x)$ with $a>1$, the graph increases slowly from left to right and passes through the point $(1,0)$ because $\log_a(1)=0$. It has a vertical asymptote at $x=0$, which means the graph gets closer and closer to the $y$-axis but never touches it.

If $0<a<1$, then the graph is decreasing. This is because the inverse of a decreasing exponential function is also decreasing. For example, $y=\log_{\frac12}(x)$ decreases as $x$ increases.

Transformations are also important. If $y=\log_a(x-h)+k$, then the graph is shifted right by $h$ units and up by $k$ units. If $y=\log_a(-x)$, the graph is reflected in the $y$-axis, but the domain changes to $x<0$. If a coefficient multiplies the function, such as $y=c\log_a(x)$, the graph is stretched or compressed vertically.

Example: consider $y=\ln(x-2)+1$. The basic graph of $y=\ln(x)$ is shifted right 2 units and up 1 unit. The domain becomes $x>2$, and the vertical asymptote becomes $x=2$. A quick point check helps: when $x=3$, $y=\ln(1)+1=1$.

Laws of logarithms and simplifying expressions

Logarithmic laws are essential tools for simplifying and solving expressions. They come directly from exponential laws.

The main laws are:

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

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

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

These work only when $M>0$ and $N>0$. students, this positivity condition matters because logarithms are only defined for positive arguments.

Example: simplify $\log_3(27x^2)$.

Since $27=3^3$, we have

$$\log_3(27x^2)=\log_3(27)+\log_3(x^2)=3+2\log_3(x)$$

provided $x>0$.

Another useful identity is change of base:

$$\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$$

This is useful when a calculator only has $\log$ or $\ln$. For example,

$$\log_2(7)=\frac{\ln(7)}{\ln(2)}$$

This formula is often needed in IB exam questions when the base is not convenient.

Solving logarithmic equations and inequalities

To solve a logarithmic equation, first use logarithm laws to combine terms if needed, then rewrite the equation in exponential form. Always check your answers at the end, because logarithmic equations can produce extraneous solutions.

Example 1: solve $\log_2(x)=5$.

Rewrite in exponential form:

$$x=2^5$$

So $x=32$.

Example 2: solve $\ln(x)+\ln(x-3)=\ln(10)$.

First combine the logs:

$$\ln\big(x(x-3)\big)=\ln(10)$$

$$x(x-3)=10$$

which gives

$$x^2-3x-10=0$$

Factor:

$$(x-5)(x+2)=0$$

So $x=5$ or $x=-2$. But the domain requires $x>0$ and $x-3>0$, so $x>3$. Therefore only $x=5$ is valid.

For inequalities, the key idea is that logarithms preserve order only when the base is greater than $1$. For example, if $a>1$, then $\log_a(x)$ is increasing. So

$$\log_a(u) > \log_a(v)$$

gives

$$u>v$$

as long as $u>0$ and $v>0$. If $0<a<1$, the inequality reverses.

Example: solve $\log_3(x-1)\le 2$.

Because base $3>1$, rewrite as

$$x-1\le 3^2$$

$$x\le 10$$

Also, the domain requires $x-1>0$, so $x>1$. The solution is

$$1<x\le 10$$

This combination of algebra and domain checking is a very important IB skill.

Connections to exponential models and real-world use

Logarithmic functions often appear as the inverse of exponential growth models. If a quantity grows according to $y=ab^t$, then solving for time $t$ usually requires logarithms. For example, if a population follows

$$P=500(1.08)^t$$

and you want to know when $P=1000$, solve

$$1000=500(1.08)^t$$

Divide by $500$:

$$2=(1.08)^t$$

Now take logs:

$$t=\frac{\ln(2)}{\ln(1.08)}$$

This gives the time needed for the population to double.

Logarithms are also useful in models where change is rapid at first and then slows down. Because logarithmic growth increases slowly, it can describe learning curves, certain biological measurements, and some data trends where early changes are large but later changes are smaller. 📊

In IB Mathematics: Analysis and Approaches HL, logarithmic functions connect to many other topics: function transformations, inverses, composite functions, solving equations, inequalities, and modeling. They also support reasoning about graphs and how functions behave near asymptotes.

Conclusion

Logarithmic functions are a reverse language for exponents. students, when you understand that $\log_a(b)=x$ means $a^x=b$, the rest becomes much more manageable. You can interpret graphs, apply laws, solve equations, and analyze inequalities with confidence. Since logarithms are inverses of exponential functions, they fit naturally into the IB study of functions and provide a powerful tool for modeling and problem-solving. Keep checking domains, using laws carefully, and rewriting expressions in the most useful form. With practice, logarithms become a flexible and reliable part of your mathematics toolkit ✅

Study Notes

A logarithm answers the question “to what power?”: $\log_a(b)=x$ means $a^x=b$.
The base must satisfy $a>0$ and $a\ne 1$; the argument must satisfy $b>0$.
Common logarithm: $\log(x)$ has base $10$.
Natural logarithm: $\ln(x)$ has base $e$.
Logarithmic functions are inverses of exponential functions, so their graphs reflect in the line $y=x$.
For $y=\log_a(x)$ with $a>1$, the graph is increasing and has vertical asymptote $x=0$.
The point $(1,0)$ lies on every graph of $y=\log_a(x)$ because $\log_a(1)=0$.
Log laws: $\log_a(MN)=\log_a(M)+\log_a(N)$, $\log_a\left(\frac{M}{N}\right)=\log_a(M)-\log_a(N)$, and $\log_a(M^k)=k\log_a(M)$.
Change of base: $\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$.
When solving logarithmic equations, always check the domain to avoid extraneous solutions.
For inequalities, remember that the direction depends on the base: it stays the same if $a>1$ and reverses if $0<a<1$.
Logarithms are useful in exponential growth models because they help solve for the exponent variable.
In IB Functions, logarithms connect graphing, inverses, transformations, and equation solving.