Solve for x 10^(2x)+2(10^x)-24=0

Problem

10^(2*x)+2*(10^x)−24=0

Solution

1. Substitute a new variable to simplify the equation by letting u=10^x

2. Rewrite the equation in terms of u noting that 10^(2*x)=(10^x)^2=u^2

u^2+2*u−24=0

3. Factor the quadratic equation by finding two numbers that multiply to −24 and add to 2

(u+6)*(u−4)=0

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

u=−6

u=4

5. Substitute back 10^x for u and solve for x

10^x=−6

10^x=4

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

7. Apply logarithms to the valid equation 10^x=4

(log_10)(10^x)=(log_10)(4)

x=(log_10)(4)

Final Answer

x=(log_10)(4)

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