Solve for x 3^(2x)+3^x-6=0

Problem

3^(2*x)+3^x−6=0

Solution

1. Rewrite the equation using the property of exponents (a^b)^c=a^(b*c) to make it a quadratic form.

(3^x)^2+3^x−6=0

2. Substitute a new variable u=3^x to simplify the equation into a standard quadratic.

u^2+u−6=0

3. Factor the quadratic equation by finding two numbers that multiply to −6 and add to 1

(u+3)*(u−2)=0

4. Solve for u by setting each factor to zero.

u=−3

u=2

5. Back-substitute 3^x for u and solve for x

3^x=−3

3^x=2

6. Evaluate the results. Since 3^x must always be positive for real x the equation 3^x=−3 has no real solution. Solve 3^x=2 using logarithms.

ln(3^x)=ln(2)

x*ln(3)=ln(2)

x=ln(2)/ln(3)

Final Answer

x=ln(2)/ln(3)

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