Inverse Functions and the Logarithm

Inverse Function

If $f$ is a function, then the function $g$ is the inverse function of $f$ if

$g(f(x))=x$

for all $x$ in which $f(x)$ can be calculated.

Details

The inverse of a function $f$ is denoted by $f^{-1}$, i.e.

$f^{-1}(f(x))=x$

Examples

Example

If $f(x) = x^2$ for $x<0$ then the function $g$, defined as $g(y)=\sqrt{y}$ for $y>0$, is not the inverse of $f$ since $g(f(x))=\sqrt{x^2} = |x| = -x$ for $x<0$.

When the Inverse Exists: The Domain Question

Inverses do not always exist. For an inverse of $f$ to exist, $f$ must be one-to-one, i.e. for each $x$, $f(x)$ must be unique.

Figure: The function $f(x)=x^2$ does not have an inverse since $f(x)=1$

has two possible solutions $-1$ and $1$.

Examples

Example

$f(x)=x^2$ does not have an inverse since $f(x)=1$ has two possible solutions -1 and 1.

Note

Note that iff $f$ is a function, then the function $g$ is the inverse function of $f$, if $g(f(x)) = x$ for all calculated values of $x$ in $f(x)$.

The inverse function of $f$ is denoted by $f^{-1}$, i.e. $f^{-1}(f(x)) = x$.

Example

What is the inverse function, $f^{-1}$, of $f$ if $f(x) = 5 + 4x$. The simplest approach is to write $y=f(x)$ and solve for $x$.

With

$f(x) = 5 + 4x$

we write

$y = 5 + 4x$

which we can now rewrite as

$y - 5 = 4x$

and this implies

$\displaystyle\frac{y-5}{4} = x$

And there we have it, very simple:

$f^{-1}(f(x)) = \displaystyle\frac{y - 5}{4}$

The Base 10 Logarithm

When $x$ is a positive real number in $x=10^y$, $y$ is referred to as the base 10 logarithm of x and is written as:

$y=\log_{10}(x)$

or

$y=\log(x)$

Details

If $\log (x) = a$ and $\log (y)=b$, then $x = 10^a$ and $y = 10^b$, and

$x \cdot y = 10^a \cdot 10^b = 10^{a+b}$

so that

$\log(xy) = a+b$

Examples

Example

$$\begin{aligned} \log(100) &= 2 \\ \log(1000) &= 3 \end{aligned}$$

Example

If

$\log(2) \approx 0.3$

then

$10^y=2$

Note

$2^{10}=1024 \approx 1000 = 10^3$

therefore

$2 \approx 10^{3/10}$

so

$\log (2) \approx 0.3$

The Natural Logarithm

A logarithm with $e$ as a base is referred to as the natural logarithm and is denoted as $\ln$:

$y=\ln(x)$

if

$x=e^y=\exp(y)$

Note that $\ln$ is the inverse of $\exp$.

Figure: The curve depicts the function $y=\ln(x)$ and shows that $\ln$ is the inverse of $\exp$. Note that $\ln(1)=0$ and when $y=0$ then $e^0=1$.

Properties of Logarithm(s)

Logarithms transform multiplicative models into additive models, i.e.

$\ln(a\cdot b) = \ln a + \ln b$

Details

This implies that any statistical model, which is multiplicative becomes additive on a log scale, e.g.

$y = a \cdot w^b \cdot x^c$

$\ln y = (\ln a) + \ln (w^b) + \ln (x^c)$

Next, note that

$$\begin{aligned} \ln (x^2) &= \ln (x \cdot x) \\ &= \ln x + \ln x \\ &= 2 \cdot \ln x \end{aligned}$$

and similarly $\ln (x^n) = n \cdot \ln x$ for any integer $n$.

In general $\ln (x^c) = c \cdot \ln x$ for any real number c (for $x>0$).

Thus the multiplicative model (from above)

$y=a \cdot w^b \cdot x^c$

becomes

$y= (\ln a) + b \cdot \ln w + c \cdot \ln x$

which is a linear model with parameters $(\ln a)$, $b$ and $c$.

In addition, the log-transform is often variance-stabilizing.

The Exponential Function and the Logarithm

The exponential function and the logarithms are inverses of each other

$x = e^y \leftrightarrow y = \ln{x}$

Details

Note

Note the properties:

$\ln (x \cdot y) = \ln (x) + \ln (y)$

and

$e^a \cdot e^b = e^{a+b}$

Examples

Example

Solve the equation

$10e^{1/3x} + 3 = 24$

for $x$.

First, get the $3$ out of the way:

$10e^{1/3x} = 21$

Then the $10$:

$e^{1/3x} = 2.1$

Next, we can take the natural log of 2.1. Since $\ln$ is an inverse function of $e$ this would result in

$\displaystyle\frac{1}{3}x = \ln(2.1)$

This yields

$x = \ln(2.1) \cdot 3$

which is

$\approx 2.23$