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 1–4.3(2)^(x-2) + 1 = 100

Verified step by step guidance

1. Step 1: Start by isolating the exponential term. Subtract 1 from both sides of the equation to get 3(2)^{x-2} = 99.
2. Step 2: Divide both sides by 3 to further isolate the exponential term, resulting in (2)^{x-2} = 33.
3. Step 3: Apply the logarithm to both sides to solve for the exponent. You can use the natural logarithm (ln) or common logarithm (log). For example, ln((2)^{x-2}) = ln(33).
4. Step 4: Use the logarithmic identity ln(a^b) = b*ln(a) to bring down the exponent: (x-2) * ln(2) = ln(33).
5. Step 5: Solve for x by dividing both sides by ln(2) and then adding 2 to both sides: x = (ln(33)/ln(2)) + 2.

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 exhibit rapid growth or decay, depending on the base. Understanding their properties is crucial for solving equations involving exponential terms, as they often require logarithmic manipulation to isolate the variable.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an equation. The logarithm log_b(a) answers the question: 'To what exponent must the base 'b' be raised to produce 'a'?' This concept is essential for solving exponential equations, as it enables us to transform the equation into a linear form, making it easier to isolate the variable.

Rounding and Decimal Approximation

Rounding is the process of adjusting a number to a specified degree of accuracy, often to make it simpler to work with. In this context, rounding to the nearest thousandth means keeping three decimal places. This concept is important when providing solutions in decimal form, as it ensures that the answers are both precise and manageable, particularly when dealing with irrational numbers that cannot be expressed exactly.