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)`.