Logarithmic functions

A logarithmic function is the inverse of an exponential function.

What is a logarithm

We write a logarithmic function with the word $\log$, and if it is a natural logarithm (see below), we label it ln. The basic prescription of a logarithmic function looks like this:

$$y=\log_ax$$

This notation reads "The logarithm of the number x with base a". This logarithmic function is the inverse of the exponential function f(x) = a^x. The functional value of the logarithmic function is called the logarithm.

Since the logarithmic function is the inverse of the exponential function, the following equivalence must hold:

$$\Large y=\log_ax\quad\Leftrightarrow\quad a^y=x$$

Thus, if the value of a^y is equal to x, then the logarithm of x with base a is equal to y. Let us try to show some examples. Consider the exponential equation f(x) = 2^x. What would the inverse (logarithmic) function look like? Like this: $f^{-1}(x)=\log_2x$. If we add a triple to the exponential function, we get: f(3) = 2³ = 8.

What relationship must now hold? It must hold that if we put an eight in the inverse function f⁻¹, the logarithmic function must return a three. Since the number 3 was the argument of the exponential function and the number 8 was the result. The inverse function behaves the other way around, the input is the number 8 and the output is the number 3. Meanwhile, you can check on Google that this is indeed the case (lg stands for logarithm with base two).

So if we write 2³ = 8, then the inverse logarithmic function gives us the relationship: $\log_28=3$. So what is a logarithm? The logarithm is the exponent to which we must multiply the base to get the argument x.

Important logarithmic functions

Some logarithmic functions are particularly important. In particular, the "natural logarithm", which has the Euler number as its base. We denote this by e. It is an irrational number, that is, a number with infinite decimal expansion. Its approximate value is: e = 2,718 281 828…. We write the natural logarithm either as $\log_ex$ or more simply as ln x. The letter "n" there is from the Latin "logaritmus naturalis", but for memorization you need only the English, where it is similar: "natural" = "natural".

Another important logarithm is the "decadic logarithm", which is a logarithmic function with base ten. It is usually written as either $\log_{10}x$ or just $\log x$. If no base is given for a logarithm, it is assumed to have a base of 10.

Properties of the logarithmic function

As we already know, the logarithmic function is the inverse of the exponential function. As a result, it also takes on some of its properties and limitations. We know that the exponential function has the form f(x) = a^x, where a is a real number that is greater than zero and different from one. This makes the base of the logarithm a must also be greater than zero and different from one. So once again, the notation of the logarithmic function:

$$y=\log_ax,\quad a\in\mathbb{R},\quad a>0,\quad a\ne1$$

Since it is an inverse function, then we also know the defining domain and the domain of values. The definitional domain of a logarithmic function is the same as the domain of values of an exponential function, so the definitional domain of a logarithmic function is equal to (0, ∞). The domain of values of the logarithm is then the same as the definitional domain of the exponential function, i.e. the set of all real numbers.

Graphs of logarithmic functions

Similar to the exponential function, we must distinguish between two cases. When the base a is from the interval (0, 1) and when it is from the interval (1, ∞). In the first case, i.e. a is from the interval (0, 1), the graph looks like this:

In the case when the base a is from the interval (1, ∞), the graph looks like this:

See also the graph of the exponential function and its inverse logarithmic function in one graph (blue is the logarithm, red is the exponential function):

Why do the graphs intersect at the point [1, 0]

Both graphs intersect the x axis at the point x = 1. This is fine, since every exponential function passes through the point [0, 1]. Since the logarithm is an inverse function, this function must always pass through the point [1, 0]. It makes sense. If the graph passes through the point [1, 0], it means that for the input of the function x = 1 we have the output f(x) = 0.

In the case of logarithms, this means that we are looking for the exponent by which, when we multiply the base, we get one. What is the exponent? Only zero. Anything to zero is one, so whatever the base is, the logarithmic function will pass through the point [1, 0], because anything to zero is one. (Note: remember that the base can't be anything at all: a>0 and a≠1.)

Theorems about logarithms (formulas)

The following are some important relationships and formulas we can say about logarithms:

Suppose that the base a is indeed the base of a logarithm, i.e., a>0, a≠1. Further, let x₁ and x₂ be arbitrary positive real numbers. Then:

$$\begin{eqnarray} \log_a(x_1\cdot x_2)&=&\log_a x_1+\log_a x_2\\ \log_a\left(\frac{x_1}{x_2}\right)&=&\log_a x_1 - \log_a x_2\\ \log_a x^r&=&r\cdot\log_ax\quad\forall r\in\mathbb{R}\\ \log_a\sqrt[n]{x}&=&\frac{1}{n}\log_ax\quad\forall n\in\mathbb{N} \end{eqnarray}$$

Some relations that follow directly from the definition of the logarithm:

$$\begin{eqnarray} \log_a1&=&0\quad(a^0=1)\\ \log_aa&=&1\quad(a^1=a)\\ a^{\log_a x} &=& \log_a{a^x} = x \end{eqnarray}$$

How to use the natural logarithm to express another logarithm

It sometimes happens that, for example, on a calculator, we do not have a logarithm of arbitrary base, but only natural and decadic. What to do in case you need to calculate a logarithm with a different base? There is a formula to help you. It is true that:

$$\log_ax=\frac{\log_bx}{\log_ba}$$

If we choose Euler's number as the value of b, this gives us the formula:

$$\log_ax=\frac{\ln x}{\ln a}$$

In the first chapter we needed to calculate the logarithm of the number 8 with base 2. We can use the natural logarithm to calculate this as follows:

$$\log_28=\frac{\ln8}{\ln2}=3$$

Again, we can check the calculation on Google. But you don't have to use the natural logarithm, you can use the decadic logarithm, the formula allows it. So, anyway:

$$\log_28=\frac{\log8}{\log2}=3$$

(Remember that if no base is given, the base is assumed a = 10.) Again, check on Google.

Calculator

If you need to calculate the logarithm, you can use the logarithm calculator here 🧮.