Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. (1/3)^x = -3

Verified step by step guidance

1. Recognize that the equation is \(\left(\frac{1}{3}\right)^x = -3\). Here, the base \(\frac{1}{3}\) is a positive number less than 1, and the expression represents an exponential function.
2. Recall that an exponential function with a positive base (other than 1) raised to any real power will always yield a positive result. This means \(\left(\frac{1}{3}\right)^x > 0\) for all real values of \(x\).
3. Since the right side of the equation is \(-3\), which is negative, there is no real value of \(x\) that can satisfy the equation \(\left(\frac{1}{3}\right)^x = -3\).
4. Conclude that the equation has no real solutions because an exponential function with a positive base cannot equal a negative number.
5. If complex solutions are considered, you would need to use logarithms with complex numbers, but since this is a College Algebra problem focusing on real solutions, the answer is that no real solution exists.

Key Concepts

Properties of Exponential Functions

Exponential functions have the form a^x where the base a is positive and not equal to 1. Their outputs are always positive numbers, meaning expressions like (1/3)^x cannot equal a negative number. Understanding this helps identify when an equation has no real solution.

Domain and Range of Exponential Functions

The domain of an exponential function is all real numbers, but its range is strictly positive real numbers (0, ∞). This means exponential expressions cannot produce zero or negative values, which is crucial when solving equations involving exponentials.

Solving Exponential Equations

To solve exponential equations, isolate the exponential expression and apply logarithms if necessary. However, if the equation sets an exponential equal to a negative number, no real solution exists because exponentials never output negatives.