Solve each exponential equation. Express the solution set so...

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator.
$$\left(\frac{1}{3}\right)^{x}=-3$$

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Key Concepts

Domain and Range Considerations

Understanding the domain and range of functions helps in assessing whether a given equation is solvable. For exponential functions with a positive base, the range is strictly positive. Therefore, any equation that sets such an exponential expression equal to a negative number has no solution in the real number system.

Properties of Exponential Functions

Exponential functions with a positive base always yield positive outputs for any real exponent. This means that regardless of the value of the exponent, the function will never take on a negative value. Recognizing this property is crucial when determining the viability of an equation's proposed solutions.

Exponential Equations

Exponential equations involve expressions where the variable appears in the exponent. Solving these equations often requires the use of logarithms to 'bring down' the exponent or utilizing properties of exponential functions, but it is essential to consider the inherent properties of the exponential function before proceeding.

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