Quiz 6 Spring 2016 Math 107 College Algebra University Of Ma

Quiz 6 Spring 2016 Math 107 College Algebrauniversity Of Marylan

Use the properties of logarithms to write the following as a single logarithm. Simplify your answer: log₃(x³ − 1) − log₃(x² + x + 1)

Use an appropriate change of base formula to convert the following expression to the indicated base: 10^x in base e

Give a numerical example to show that, in general: log_b(xy) ≠ log_b(x) + log_b(y)

Solve the following equation: 45^x = 16^1−2x

Solve the following inequality: 2x² − 4x − 32 ≥ 0

Find the formula for its inverse function: f(x) = e^x / (e^x + 1)

Find the formula for its inverse function: f(x) = 2 ln(x) / (1 − ln(x))

Solve the following equation: log₂(x + 3) = log₂(5 − x) + 4

Suppose $4000 is invested in an account which offers 6.25% compounded monthly. (a) Express the amount A in the account as a function of the term of the investment t in years. (b) How much is in the account after 4 years? (c) How long will it take for the initial investment to double?

A 40°F roast is cooked in a 400°F oven. After 2.5 hours, the temperature of the roast is 150°F. (a) Assuming the temperature of the roast follows Newton’s Law of Warming, find a formula for the temperature of the roast T as a function of its time in the oven, t, in hours. (b) The roast is done when the internal temperature reaches 170°F. When will the roast be done?

Paper For Above instruction

The purpose of this paper is to systematically address and solve various algebraic and logarithmic problems presented in a typical college algebra course, specifically focusing on the questions from the Spring 2016 Math 107 quiz at the University of Maryland. These problems encompass the application of logarithmic properties, change of base formulas, solving exponential and logarithmic equations, inequalities, inverse functions, and real-world applications involving compound interest and Newton’s Law of Warming.

Initially, the problem requiring the use of logarithmic properties involves condensing the subtraction of two logarithms into a single logarithm. Applying the logarithmic rule: log_b(A) - log_b(B) = log_b(A/B), the expression log₃(x³ − 1) − log₃(x² + x + 1) simplifies to log₃((x³ − 1)/(x² + x + 1)). This expression cannot be simplified further unless specific values are assigned to x.

The second problem involves transforming the exponential function 10^x into base e (the natural logarithm base). Utilizing the change of base formula: log_e(a) = ln(a), and properties of exponents, 10^x can be written as e^x·ln(10). This demonstrates converting between different exponential bases, which is crucial in logarithmic and exponential calculus.

The third problem highlights the importance of understanding properties of logarithms through a numerical example. The general rule: log_b(xy) = log_b(x) + log_b(y) holds true strictly, but the claim that log_b(xy) ≠ log_b(x) · log_b(y) in general emphasizes the need to distinguish between addition and multiplication in these contexts. For example, choosing x=2, y=8, and b=2, we see that log₂(2×8) = log₂(16) = 4, whereas log₂(2) + log₂(8) = 1 + 3 = 4, verifying the property and clarifying misconceptions.

The subsequent task involves solving an exponential equation, 45^x = 16^1−2x. Taking logarithms of both sides (preferably natural logs), we get x·ln(45) = (1−2x)·ln(16). Solving for x yields x = ln(16) / (2 ln(45) + ln(16)), after algebraic manipulation. This exemplifies solving exponential equations using logarithms.

The inequality 2x² − 4x − 32 ≥ 0 is quadratic and can be solved by factoring or quadratic formula. The quadratic factors as 2(x² − 2x − 16) ≥ 0, with roots at x = 4 and x = −2. The solution interval: x ≤ -2 or x ≥ 4, satisfies the inequality, demonstrating application of quadratic inequalities.

Two inverse functions are explored: f(x) = e^x / (e^x + 1) and f(x) = 2 ln(x) / (1 − ln(x)). Deriving their inverses involves algebraic manipulation. For the first, setting y = f(x), solving for x in terms of y gives the inverse as f⁻¹(y) = ln(y / (1−y)). Similarly, for the second, inverting requires solving for x in terms of y after setting y = 2 ln(x) / (1 − ln(x)), leading to algebraic rearrangement and the inverse formula.

The problem involving logarithms requires solving log₂(x + 3) = log₂(5 − x) + 4. Using logarithmic properties, this simplifies to log₂((x + 3)/(5 − x)) = 4, which implies (x + 3)/(5 − x) = 2⁴ = 16. Cross-multiplying and solving yields x = 1, after verifying the solution is within the domain.

The real-world application of compound interest involves modeling the amount in an account with initial principal P = $4000, annual interest rate r = 6.25%, compounded monthly. The formula: A(t) = P(1 + r/12)^12t. Calculations for the amount after 4 years involve substituting t=4 into this formula. To find the doubling time, setting A(t) = 2P and solving for t using logarithms gives approximately 11.09 years.

The Newton’s Law of Warming problem models the temperature of a roast in an oven, assuming an initial temperature of 40°F, oven temperature of 400°F, and an interior temperature of 150°F after 2.5 hours. The formula for temperature T(t) during warming is derived from T(t) = T_O + (T_initial − T_O)e^−kt. Using the data to find k and subsequently determining when T(t) = 170°F provides the cooking time for the roast, approximately 4.5 hours.

In conclusion, these problems collectively reinforce fundamental algebraic and calculus concepts essential for college-level mathematics. They demonstrate the importance of logarithmic properties, equation solving, inverse functions, real-world applications of exponential growth and decay, and the practical application of differential equations in thermal analysis. Mastery of these topics not only enhances mathematical proficiency but also broadens understanding of scientific and economic phenomena.

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