# Properties of Logarithms You'll Be Stunned to Know

Mathematics can be an interesting subject, if the properties of logarithms are clear to you. This write-up will help you understand these concepts better. Take a look...

Narayani Karthik

Last Updated: Jan 2, 2019

^{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.

_{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.

Properties of Natural Logarithms

The definition of a common logarithm is can be expressed as, y = log

_{a}x, 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) = log_{10}x. However, if the base is an irrational constant, which is approximately 2.718281828, then the expression is rewritten as ln (x) ~ log_{e}x.- ln (xy) = ln x + ln y
- ln (x/y) = ln x - ln y
- ln x
^{y}= y ln x - ln e
^{x}= x - e
^{ln x}= x - ln e = 1
- ln 1 = 0
- ln (1/x) = - ln (x)

- log
_{a}xy = log_{a}x + log_{a}y - log
_{a}(x/y) = log_{a}x - log_{a}y - log
_{a}x^{y}= y * log_{a}x - log
_{a}a = 1 - log
_{a}1 = 0

Inverse Properties

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_{a}b = x, where x = log_{a}b is the logarithmic form, and a^{x}= b is the exponential form. Now, the inverse logarithm properties are based on this expression.- log
_{a}a^{x}= x - a
^{logax}= x

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

^{x}

So, the inverse logarithmic expression for the above would be: f

^{-1}(x) = log

_{a}x

The general form inverse algorithm can be written as:

- f(f
^{-1}(x)) = f(log_{a}x) = a^{logax} - f
^{-1}(f(x)) = log_{a}a^{x}(i.e. f(f^{-1}(x)) = x)

*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