How do I solve (3^2x) - (3^2+x) +18 = 0

Question

Simplifying Exponents

Answer 1

Is this the same as

 

3^2x - 3^2+x + 18 = 0 ?

 

I could not be sure from what you typed into your question.

 

I will rewrite the second term on the left side:

 

3^2x - 3²3^x + 18 = 0

 

3^2x - 9·3^x + 18 = 0.

 

You may recognize that this is a quadratic equation in 3^x. Let b = 3^x. then b² = (3^x)² = 3^2x.

 

So, the quadratic equation in b is b² - 9b + 18 = 0. This can be solved by factoring:

 

(b - 6)(b - 3) = 0  ⇒  b = 3 or b = 6.

 

To finish up, replace b by 3^x in both of these and solve each for x.

 

3^x = 3  ⇒  x = 1

 

3^x = 6  ⇒  x = log₃ 6 = (log 6)/(log 3) ≈ 1.631 to three decimal places.

 

This means there are two solutions: x = 1 and x ≈ 1.631.