Question

Find the domain and solve for x in the equation (7^x - 5)^(6 - x) = 49.

A) Domain: All real numbers; x ≈ 1.165
B) Domain: All real numbers; x ≈ 1.062
C) Domain: x ≥ 0; x ≈ 1.165
D) Domain: x ≥ 0; x ≈ 1.062

Answer 1

The domain of the equation is all real numbers. The solutions for x are approximately 1.165 and 1.062.

Step-by-step explanation:

To find the domain and solve for x in the equation (7x - 5)(6 - x) = 49, we can start by finding the domain. Since we can raise any positive number to any real power, the domain is all real numbers.

Next, we can solve the equation by simplifying the expression inside the parentheses. Taking the natural logarithm of both sides, we get (x ln(7) - ln(5))(6 - x) = ln(49). Expanding and rearranging the equation, we find a quadratic equation: -(x^2 - 6x + ln(7)ln5 - ln(49)) = 0. Using the quadratic formula, we can find the solutions for x. The solutions are approximately x ≈ 1.165 and x ≈ 1.062.