Mathematics Magazine for Grades 1-12  

 

3/2004

     

I am not afraid of tomorrow, for I have seen yesterday and I love today.
 - William Allen White

   Grade 12

Theory:

RULES OF LOGARITHMS

Let a be a positive number such that a does not equal 1, let n be a real number, and let u and v be positive real numbers.

Logarithmic Rule 1: Loga (uv) = Loga(u) + Loga(v)

Logarithmic Rule 2:  

Logarithmic Rule 3:

Solutions from the Previous Issue: 

Proposed by Mihai Rosu Professor

1         Prove that the function    is a constant function

                        for every .

Solution:

        Let us calculate value of function, we have:

   

=

.

 

2         Prove the identity.

Solution: We use the equality: and

 

3         Prove the identity .

Solution: We have successfully

,

Here we used the relation .

 

4         Prove that .

Solution:

 

 

 Where we used , .

 

5         Prove that: .

Solution: We know that , then we have

.

6         Find    such as the expression   has a minim value.

Solution: Let A= .

A has a minimum if  is maximum, this is for  =1, thus  .

 

7         Solve the equation: .

Solution: We have  .

  So    .

8         Solve for x: . 

Solution:   

If      we have 

  .

So  .

9         Evaluate the trigonometric limits: .

Solution: It is known  .

We denote , if   then

10      Calculate the limit: .

Solution:

 

. We used and