How is logarithm a function?

Question

What would define logarithm as a function. I know logbase10 1000 = x means 10^x = 1000. but what is the input here? Is it log base x or logbase10 of x? Can you define the function ''logarithm'', explain clearly what it does, how it works, and so on... ? I know exponentiation is the opposite of logarithm but exponentiation is not a function. 2^3 is not a function. And if it were a function would the input be 2 or 3?

The inputs for exponentiation are both 2 and 3. $2^3$ is not a function, it's a value; but $2^x$ is a function of $x$, and $y^3$ is a function of $y$, and $y^x$ is a function of $x$ and $y$ both. For logarithms we often fix the base, and so e.g. $\log_{10}(x)$ is defined as the value $y$ such that $10^y=x$. — Steven Stadnicki, May 25 '21 at 18:05

Tristan Duquesne · Accepted Answer · 2021-05-25T19:01:14.303

The logarithm is a function which can be used to extract an exponent for a given number. $\log_b(x)$ returns "the number $n$ to which $b$ must be raised to obtain $b^n = x$". This extends to $b$ and $x$ being any positive real numbers, and $n$ any real number.

Because they are often useful, you'll see the following:

$\ln = \log_e$, the neperian logarithm. It is the standard logarithm from which we define other logarithms. $e$ is called "Euler's constant", and is one of the most important numbers in mathematics.
$\lg = \log_2$, the binary logarithm, important in algorithmics and information theory
$\log = \log_{10}$, the decimal logarithm, seen everywhere in science, since we use the metric system, which is a "powers of base ten" way of organizing values of different orders of magnitude.

Exponentiation is also very much a function. Whether you're interested in the one-input, one-output $\exp(x) = e^x$ function, which raises Euler's constant to the power $x$; or in the two-inputs, one-output general exponentiation $\text{pow}(a,b) = a^b$. Both are functions.

score 0 · Answer 2 · answered May 25 '21 at 18:09

A function like

$$f(x)=x^c$$ where $c$ is a constant is called a power function. Its inverse is called a root function (usually when $c$ is a natural, in particular $2$ and $3$), but is in fact also a power:

$$f^{-1}(x)=\sqrt[c]x=x^{1/c}.$$

A function like

$$f(x)=c^x$$ is called an exponential. Its inverse is a logarithm and is denoted

$$f^{-1}(x)=\log_c(x).$$

Thus

$$c^{\log_c(x)}=x=\log_c(c^x).$$

score 0 · Answer 3 · answered May 25 '21 at 18:19

Denote $f:]0,+\infty[ \rightarrow \mathbb{R}$ the function that satifies:

$f’(x)=\frac{1}{x}$
$f(1)=0$

I won’t really explain why such a function must exist and is unique, but that’s related to the Picard–Lindelöf theorem.

This function is noted $\ln (x)$. And it has a really interesting property: for any positive real numbers $a, b$, we have $\ln(ab)=\ln(a)+\ln(b)$. The proof is simple enough: if we consider $a$ as a variable and $b$ as a constant, then derivating the left member of the equality gives $\frac{b}{ab}=\frac{1}{a}$, which is exactly what you get by derivating the right member. And for $a=1$, both members are equal to $\ln(b)$. From this identity, you can easily show by induction that for any positive real number $a$ and any integer $n$, you have $\ln(a^n)=n \ln(a)$.

We’re almost here! We now write the function $\log_a(x)$, the logarithm base $a$, defined by $\log_a(x)=\frac{\ln (x)}{\ln (a)}$. Let’s now take your example of base $10$; we set $a=10$. Then, what is $\log_{10}(1000)$? Well, we have $\ln (1000)=\ln (10^3)=3\ln (10)$, which means $\log_{10}(1000)=3$. And that’s exactly what we were looking for!

How is logarithm a function?

3 Answers3