Logarithm

We have seen that, with a base < a = , the exponential function f ( x ) = a x is defined on R having range ( , ∞ ) . 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 < a = , we have y = a x is equivalent to log a y = x.

For example, since = we have log ( ) = . 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. (i) Note that exponential function a x is defined for all x ∈ R and a x > and so log a ( · ) defined only for positive real numbers .

(ii) Also, a = for any base a and hence log a ( ) = for any base a . Basic Algebra . . Properties of Logarithm (i) a log a x = x for all x ∈ ( , ∞ ) and log a ( a y ) = y for all y ∈ R .

(ii) For any x, y > , log a ( xy ) = log a x + log a y. (Product Rule) (iii) For any x, y > , log a x = log a x − log a y. (Quotient Rule) (iv) For any x > and r ∈ R , log a x r = r log a x. (Power Rule) (v) For any x > , with a and b as bases, log b x = log a x log a b .

(Change of base formula.) Proof. Since exponential function with base a and logarithm function with base a are inverse of each other, (i) follows by using the definitions. (ii) For x, y > let log a x = u , log a y = v , and log a ( xy ) = w . Rewriting these in the exponential form we obtain a u = x , a v = y , and, a w = xy .

So, a w = xy = a u a v = a u + v ; thus w = u + v. Thus, we obtain log a ( xy ) = log a x + log a y . (iii) Let log a x = u , log a y = v , and log a y = w . Then a u = x , a v = y and a w = x y .

Hence, a w = x y = a u a v = a u − v ; which implies w = u − v . Thus, we obtain log a x = log a x − log a y. (iv) Let log a x = u . Then a u = x and therefore, x r = ( a u ) r = a ru .

Thus, log a x r = ru = r log a x . (v) Let log b x = v. We have b v = x . Taking logarithm with base a on both sides we get log a b v = log a x .

On the other hand log a b v = v log a b by the Power rule. Therefore, v log a b = log a x. Hence log b x = log a x log a b , b > . This completes the proof.

Remark: (i) If a = , then the corresponding logarithmic function log x is called the common logarithm . (ii) If a = e ,(an irrational number, approximately equal to . ), then the corresponding logarithmic function log e x is called the natural logarithm . It is denoted by ln x.

These above particular cases of logarithmic functions are used very much in other sciences and engineering. Particularly, the natural logarithm occurs very naturally. When we write log x we mean log e x . (iii) If a = , then the corresponding logarithmic function log x called the binary logarithm, which is used in computer science.

(iv) Observe that log a log a ( ∗ ) log a + log a ; log a log a − log a . log a x = x log a ; log = log log . (v) Observe the graph of the logarithmic and exponential functions. .

Logarithm – – – – y = log x y = x – Figure . Example . Find the logarithm of to the base . Let log = x .

Then we have ( ) x = = = ( ) . Hence, ( ) x = ( ) . Therefore x = . That is, log = .

Example . If the logarithm of to base a is , then find a . We are given log a = , which gives a = = ( ) . Therefore a = .