SOLUTION: Find the real values of x which satisfy the equation. Log[base3]x+log[basex]3=10/3

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Find the real values of x which satisfy the equation. Log[base3]x+log[basex]3=10/3      Log On


   

Question 985654: Find the real values of x which satisfy the equation.
Log[base3]x+log[basex]3=10/3
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Log[base3]x+log[basex]3=10/3
log%283%2C%28x%29%29%2Blog%28x%2C3%29%22%22=%22%2210%2F3

Use the base-swapping formula log%28A%2C%28B%29%29=1%2Flog%28B%2C%28A%29%29
on the second term on the left:

log%283%2C%28x%29%29%2B1%2Flog%283%2C%28x%29%29%22%22=%22%2210%2F3

Let u = log3(x)

u%2B1%2Fu%22%22=%22%2210%2F3

Multiply through by LCD = 3u

      3uČ + 3 = 10u

3uČ - 10u + 3 = 0

  (u-3)(3u-1) = 0

u-3=0;  3u-1=0
  u=3;    3u=1
           u=1%2F3

Substitute u = log3(x)

log3(x) = 3;   log3(x) = 1%2F3

  x = 33       x = matrix%282%2C1%2C%22%22%2C3%5E%281%2F3%29%29
                   
  x = 27        x = root%283%2C3%29

Edwin