We also observed that f(x) is a bijection, hence it has an inverse. We call this inverse function as logarithmic function and is denoted by loga(.).

**Logarithm**

We have seen that,
with a base 0 *< a* ¹ 1, the exponential function *f*(*x*) = *a*^{x} is defined on R having range (0*,* *∞*)*.* We also observed that *f*(*x*) is a bijection, hence it has an inverse. We
call this inverse function as *logarithmic function* and is denoted by log_{a}(*.*). Let us discuss this
function further. Note that if *f*(*x*)
takes *x* to *y* = *a*^{x}, then log_{a}(*.*) takes *y* to *x*. That is, for 0 *< a* ¹
1*,* we have

y
= a^{x} is equivalent to log_{a} y = x.

For example, since 3^{4} = 81 we have log_{3}(81) = 4*.* In other words, with
fixed *a*, given a real number *y*, logarithm finds the
exponent *x* satisfying *a*^{x} = *y.* This is useful in
addressing practical problems like, “how long will it take for certain
investment to reach a fixed amount?” Logarithm is also very useful in
multiplying very small or big numbers.

**Properties of
Logarithm**

** **

Tags : Definition, Properties, Proof, Solved Example Problems, Exercise | Mathematics , 11th Mathematics : UNIT 2 : Basic Algebra

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11th Mathematics : UNIT 2 : Basic Algebra : Logarithm | Definition, Properties, Proof, Solved Example Problems, Exercise | Mathematics