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. 2(1.05)^x + 3 = 10

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. These functions model growth or decay processes and are characterized by their rapid increase or decrease. Understanding how to manipulate and solve equations involving exponential functions is crucial for finding solutions in problems like the one presented.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form b^y = x. The logarithm log_b(x) answers the question: to what power must the base 'b' be raised to obtain 'x'? In solving exponential equations, converting the equation to logarithmic form is often necessary to isolate the variable and find its value.

Rounding and Decimal Approximation

Rounding is the process of adjusting the digits of a number to make it simpler while maintaining its value close to the original. In this context, providing irrational solutions as decimals correct to the nearest thousandth means calculating the solution accurately and then rounding it to three decimal places. This skill is essential for presenting solutions in a clear and standardized format, especially when dealing with irrational numbers.