Expert: Paul Klarreich - 1/31/2010

Question
Solve for x over the set of real numbers:

log base(4)of x + log base(x)of 32 = 19/6.

I have attempted a solution using simultaneous equations and the change of base formula.

Answer
Questioner: Steven
Country: United States
Category: Advanced Math
Private: No
Subject: Advanced Algebra - logarithmic equation
Question: Solve for x over the set of real numbers:

log base(4)of x + log base(x)of 32 = 19/6.

I have attempted a solution using simultaneous equations and the change of base formula
................................................................

Is this what you mean by the change of base formula ??

log(base A) B * log(base B) A = 1, or
                    1
log(base A) B = ---------------
               log(base B) A

Apply that to your first term:

                      1
log (base 4) x = ----------------
                 log (base x) 4

Apply a power rule:

                        1
log(base 4) x = ------------------
                 2 log(base x) 2

and also to the second term:

log(base x) 32 = 5 log(base x) 2

Now let  y = log(base x) 2, and your equation:

log base(4)of x + log base(x)of 32 = 19/6

turns into:

 1          19
---- + 5y = ---
2y          6

You can solve it from here.