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. See Examples 1–4. (1/3)^x = -3

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay and are defined for all real numbers. Understanding their properties, such as the behavior of the function as x approaches positive or negative infinity, is crucial for solving equations involving exponents.

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction, meaning they cannot be represented as the ratio of two integers. Examples include numbers like √2 and π. When solving equations, especially those involving roots or exponents, recognizing and approximating irrational solutions to a specified decimal place is essential.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form a^x = b. The logarithm log_a(b) answers the question: 'To what power must 'a' be raised to obtain 'b'?' Understanding how to manipulate logarithmic expressions is vital for solving exponential equations, especially when dealing with non-integer solutions.