SOLUTION: find the exact value(s) of x such that 9^x-1-3^x-1-2=0

Question 214854: find the exact value(s) of x such that 9^x-1-3^x-1-2=0

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

Rewrite the left side using the identity

Simplify

Multiply EVERY term by the LCD 9 to clear out the fractions.

Simplify

Rewrite 9 as

Multiply the exponents.

Factor out the exponent 2 using the identity

Now let's make a substitution to make things easier on us. Let (since there are 2 copies of )

Replace each with 'z'

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

Rewrite as

Add to 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 solutions are or

Recall that we let . Since for all 'x', the solution isn't possible (since )

So we're only going to focus on

Start with the given equation.

Plug in

Convert to logarithmic form.

====================================
Answer:

So the only solution is

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