Expert: Sherry Wallin - 12/7/2010

Question
Sherry, I have previously asked you a question and you
helped me so much and explained each step in the problem so
well that I wanted your feedback on my final review for a
college pre-calculus class. I want to compare work and
answers with an expert.

1. Simplify, (2 + 2i) – (3 + 2i)   

2. Find the remainder of x2 - 4x + 4 if its divided by x + 2

3. Simplify, (3 – i)/(2 + i)   

4. Find the Vertical Asymptote of Y = 4 / (x - 3)   

5. Does y = (x – 3)2 + 4 cross the x-axis? yes or no.   

6. Find x given Logx25= 2

7. Simplify, Log2 132   

8. Expand, Log X^3Y^2   

9. What is the inverse of a logarithm?   

10. Find x given 2x + 2 = 2^12    

Answer
Hi Page~

Please next time show me your work and answers. I don't typically give answers.

1) -1  add the real parts and add the imaginary parts 2-3 + 2i-2i = -1

2) -2|1  -4  4
       -2  12
   --------- => x - 6 + 16/x+2  is the answer. Try (x+2)(x-6) + 16  
    1  -6 16
= x^2 -4x -12 + 16 = x^2 -4x + 4  using synthetic division

3)(3-i)/(2+i) multiply top & bottom by the conjugate of 2+i, that is multiply top and bottom by 2-i getting [3(2-i)-i(2-i)]/[2(2-i)+i(2-i)
= (6-3i-2i+i^2)/(4 -2i +2i -i^2)= (6-5i -1)/(4-(-1)) = (5-5i)/5 = 1-5i

4) The vertical asymptote is always the value that makes the denominator 0 so set x-3 = 0 and solve for x getting x = 3

5) y = (x – 3)2 + 4 => y = 2x-6+4 => y = 2x -2.  When y = 0 the line y = 2x-2 crosses the x axis so 0 = 2x-2 => 0 = 2(x-1) => 2 = 0 or x-1 = 0
=> x = 1 which is where y = 0 thus it is where the line crosses the x axis

6) Logx25= 2 => x^2 = 25 => x = +-5 but only x = 5 since the bases must not be negative

7) Log2 132 => 2^y = 132 and there is no integer y such that 2^y = 132
Notice that 2^7 = 128 and 2^8 = 256 so y has to be some number between 7 and 8. You could use a calculator that lets you do logs in any base or use the change of base formula that says
Log(base a) M = Log(base 10)M/log(base 10)a =>
log(base 10)132 = log(base 10)2 = 7.044394119

8) I am assuming this is log base 10:
Log X^3Y^2 => log(x^(6y)) = 6y*logx

9) What is the inverse of a logarithm? The exponential function

10) 2x + 2 = 2^12. There is a factor of x in every terms so cancel one factor of x giving you x + 1 = 2^11=> x = 2^11-1 = 2047

Math Prof