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. e^(x^2) = 100

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. In this context, e^(x^2) represents an exponential function where the base is Euler's number 'e', approximately equal to 2.718. Understanding the properties of exponential functions, such as their growth behavior and how to manipulate them, is crucial for solving the given equation.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the inverse function of the exponential function with base 'e'. It is used to solve equations where the variable is in the exponent. In the equation e^(x^2) = 100, applying the natural logarithm allows us to isolate the exponent, transforming the equation into x^2 = ln(100), which can then be solved for 'x'.

Irrational Numbers and Decimal Approximation

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning their decimal representation is non-repeating and non-terminating. When solving equations that yield irrational solutions, it is often necessary to provide these solutions as decimal approximations. In this case, the instruction to give irrational solutions correct to the nearest thousandth requires understanding how to round decimal values accurately to meet specified precision.