# Taking Logarithm and Exponentiating

If some expression A is formed from positive numbers with help of operations of multiplication, division and raising to power, then, using properties of logarithms, we can express log_a(A) in terms of logarithms of numbers that expression A contains. Such transformation is called taking logarithm.

Example 1. Take logarithm with base 5 of the expression (125 a^3 b^2)/(sqrt(c)), where a,b,c are positive numbers.

Using properties of logarithms we obtain that log_5((125a^3b^2)/(sqrt(c)))=log_5(125a^3b^2)-log_5(sqrt(c))=log_5(125)+log_5(a^3)+log_5(b^2)-log_5(c^(1/2))=

=3+3log_5(a)+2log_5(b)-1/2log_5(c).

Very often we need to solve inverse task: knowing logarithm of expression, we need to find expression. Such transformation is called exponentiating.

Example 2. Find x if log_3(x)=2log_3(5)+1/2log_3(8)-3log_3(10).

We have that log_3(x)=log_3(5^2)+log_3(8^(1/2))-log_3(10^3)=log_3(25)+log_3(2sqrt(2))-log_3(1000)=log_3((25*2sqrt(2))/1000)=

=log_3((sqrt(2))/(20)).

Now from equality log_3(x)=log_3((sqrt(2))/(20)) we obtain that x=(sqrt(2))/(20).