Solve for x 3^(2x)+3^(x+1)-70=0

Problem

3^(2*x)+3^(x+1)−70=0

Solution

1. Rewrite the terms using exponent rules to isolate 3^x

3^(2*x)=(3^x)^2

3^(x+1)=3^1⋅3^x=3*(3^x)

2. Substitute a new variable u for 3^x to create a quadratic equation.

u=3^x

u^2+3*u−70=0

3. Factor the quadratic equation by finding two numbers that multiply to −70 and add to 3

(u+10)*(u−7)=0

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

u=−10

u=7

5. Substitute back 3^x for u and solve for x

3^x=−10

3^x=7

6. Evaluate the solutions. Since 3^x must be positive for any real x 3^x=−10 has no real solution.

x=(log_3)(7)

7. Convert to natural logarithms using the change of base formula if needed.

x=ln(7)/ln(3)

Final Answer

x=(log_3)(7)

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