SOLUTION: please help me solve the problem log(b^2)(x)+log(x^2)b=1, b>0, b not equal to 1, x not equal to 1. Sove for x. Please note b^2 and x^2 are subscripts (bases).

Question 242901: please help me solve the problem log(b^2)(x)+log(x^2)b=1, b>0, b not equal to 1, x not equal to 1. Sove for x. Please note b^2 and x^2 are subscripts (bases).

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Use the change of base formula.

Pull down the exponents using the identity

Pull out the constants.

Factor out the GCF

Multiply both sides by 2.

Now let . Take note that . In other words,

Make the proper substitutions.

Multiply EVERY term by the LCD 'z' to clear out the fraction.

Subtract 2z from both sides.

Notice that the quadratic is in the form of where , , and

Let's use the quadratic formula to solve for "z":

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

So the only solution in terms of z is

But remember, we let

Go back to that previous equation.

Plug in

Multiply both sides by .

Since the bases are equal, the arguments are equal.

So the solution is

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