Properties of Logarithms

If x = y^b, then y = log_b (x)

As seen from this example, this is a logarithmic expression, which many of us might have come across in our academic curriculum. But what does this expression mean? It means, if y, raised to b = x, then the logarithmic value of x to the base b should be equal to y. This is one of the fundamental properties of logarithms.

The logarithm of x, to a base b, is written as log_b (x), where x is the number, base is b, and y is an exponent. Substitute numbers in this expression, say x = 100 and base b is 10, so the value of y is 2.

Logarithms were first invented by John Napier (in 1614) and Joost Burgi (in 1620). The purpose of this aspect of mathematics was to simplify very complicated mathematical expressions. Napier had an algebraic approach towards this subject, whereas Burgi's approach was geometric.

Then mathematicians like John Wallis (in 1685) and Johann Bernoulli (in 1694) came up with the idea of defining logarithms as exponents. And today, be it any field, finance or astronomy, logarithms have a major role to play in problem-solving of knotty mathematical equations.

The definition of a common logarithm is can be expressed as, y = log_ax, which holds true only if, x = a^y and a > 0. So when we say log (x), the base is implicitly 10, implying log(x) = log₁₀x. However, if the base is an irrational constant, which is approximately 2.718281828, then the expression is rewritten as ln (x) ~ log_ex.

This logarithm is often referred to as a natural logarithm. Natural logarithms are defined for all positive real numbers x, for an area defined under a curve y (covered from 1/t to 1/x). For real variables, this is also considered as real value function, and is often known as the inverse function of exponential functions.

Some of the properties, for this kind are listed here (for two variables x and y, with base e):

1. ln (xy) = ln x + ln y
2. ln (x/y) = ln x - ln y
3. ln x^y = y ln x
4. ln e^x = x
5. e^{ln x} = x
6. ln e = 1
7. ln 1 = 0
8. ln (1/x) = - ln (x)

Properties of a normal logarithm are also same, except the base is 10 for the normal one.

1. log_a xy = log_ax + log_ay
2. log_a (x/y) = log_ax - log_ay
3. log_ax^y = y * log_ax
4. log_aa = 1
5. log_a1 = 0

Consider a positive real number, which is not equal to 1, and is expressed as a^x = b. Here, x can be defined as a logarithm of b, to base a. The expression becomes log_ab = x, where x = log_ab is the logarithmic form, and a^x = b is the exponential form. Now, the inverse logarithm properties are based on this expression.

• log_aa^x = x
• a^log_ax = x

So when evaluating this, consider that the exponential function is defined as:

f(x), where f(x) = a^x

So, the inverse logarithmic expression for the above would be: f^-1(x) = log_ax

The general form inverse algorithm can be written as:

• f(f^-1(x)) = f(log_ax) = a^log_ax
• f^-1(f(x)) = log_aa^x (i.e. f(f^-1(x)) = x)

Here, let's state an example:

Question:
Determine an inverse log function of f(x) = log (x+4)

Answer:
As per the inverse properties of logarithms, f(f^-1(x)) = x.
So,

f(f^-1 (x+5)) = log (f^-1(x) + 5) = x
=> 10^x = f^-1(x) + 5
=> f^-1(x) = 10^x - 5

Logarithms are an essential aspect of math, which help in solving a lot of complex equations on a big scale. Precisely, wherever exponentials are involved, logarithms have a stellar role to play. Many complex algebraic equations are solved via logarithms, using a scientific calculator.

Not just in math, but this feature of math is used in other subjects as well, like physics, chemistry, biology, and astronomy. Even many computer based algorithms use this to solve major difficult mathematical equations.