SOLUTION: Please solve this equantion (squareroot of 1995).x^log1995(x)=x^2, find the value of x. Please note that 1995 in log1995 is subscript (base)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please solve this equantion (squareroot of 1995).x^log1995(x)=x^2, find the value of x. Please note that 1995 in log1995 is subscript (base)      Log On


   

Question 242903: Please solve this equantion
(squareroot of 1995).x^log1995(x)=x^2, find the value of x.
Please note that 1995 in log1995 is subscript (base)
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
Let me start by saying that this is one INCREDIBLE PROBLEM. I'm not sure if there is an easier way to solve this, but I solved it algebraically by taking the log base 1995 of each side. I also solved it by graphing calculator, and it came out to the SAME two answers! It must be right!!

sqrt%281995%29%2Ax%5E%28log%281995%2Cx%29%29=x%5E2

Take the log bases 1995 of each side:
log%281995%2C+sqrt%281995%29%2Ax%5E%28log%281995%2Cx%29%29%29=log%281995%2Cx%5E2%29+


By the Laws of Logarithms:

%281%2F2%29%2B+%28log%281995%2Cx%29%29%5E2=2%2Alog%281995%2Cx%29+

This is actually a quadratic equation. Start by clearing the fraction. Multiply both sides by 2:
1+%2B+2%2A%28log%281995%2Cx%29%29%5E2+-+4%2Alog%281995%2Cx%29=0+
+2%2A%28log%281995%2Cx%29%29%5E2+-+4%2Alog%281995%2Cx%29%2B1=0+

Make a substitution. Let u=log%281995%2Cx%29
2u%5E2+-4u%2B1=0

Solve this by quadratic equation:
u=%284%2B-sqrt%2816-8%29%29%2F4
u=%284%2B-2sqrt%282%29%29%2F4
u=%282%2B-sqrt%282%29%29%2F2

Now, u=log%281995%2Cx%29=%282%2B-sqrt%282%29%29%2F2
This means that 1995%5E%28%282%2B-sqrt%282%29%29%2F2%29=x

This answer was probably really hard to read in the algebra.com format, so let's just write it as
x=+1995%5Eu, where u=%28%282%2B-sqrt%282%29%29%2F2%29

x=1995%5E%28%282%2Bsqrt%282%29%29%2F2%29 or x=1995%5E%28%282-sqrt%282%29%29%2F2%29+

I calculated the values of these to be x=429887.58 and x=9.2582925 approximately. Using techniques of solving equations using graphing calculator with a TI84, I obtained exactly the same two answers.

NICE PROBLEM!! Where in the world did you get this one???

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus