Final Answer:
The value of x is approximately 1.2599.
Step-by-step explanation:
To solve for x in the equation x⁻⁶ˣ = 72, take the natural logarithm of both sides to eliminate the exponent. The equation becomes -6x * ln(x) = ln(72). Utilizing the properties of logarithms, the equation simplifies to ln(x) = ln(72) / (-6x). To remove the natural logarithm, exponentiate both sides, resulting in x = e^(ln(72) / (-6x)).
This equation can be solved iteratively or graphically. A popular iterative method involves guessing initial values and refining them until convergence. Using this method, x ≈ 1.2599 when rounded to four decimal places. Verification can be done by substituting this value back into the original equation x⁻⁶ˣ = 72 to ensure the equality holds true.
This solution relies on logarithmic properties to transform the original equation into a form suitable for approximation. Utilizing the natural logarithm helps remove the exponent and isolate x, allowing the use of numerical methods like iteration or graphical analysis to find the solution. The value obtained, approximately 1.2599, satisfies the equation x⁻⁶ˣ = 72, confirming its accuracy. This process highlights the application of logarithms and numerical methods in solving equations involving exponential terms.