Unit 7: Exponential and Logarithmic Functions – Homework 4 Answers
The focus of Unit 7 is on the behavior of exponential and logarithmic functions, the rules that govern them, and their real‑world applications. Homework 4 tests the ability to manipulate these functions, solve equations, and interpret graphs. Below is a comprehensive walkthrough of each problem, the underlying concepts, and the final answers.
Introduction
Understanding exponential and logarithmic functions is essential for fields ranging from finance to biology. In this guide, we’ll dissect every question from Homework 4, showing the step‑by‑step logic that leads to the correct answer. By the end, you should feel confident tackling similar problems on your own And it works..
Problem 1 – Solving an Exponential Equation
Question:
[ 3^{2x+1}=81 ]
Solution Steps
-
Recognize a common base
(81) can be expressed as (3^4).
[ 3^{2x+1}=3^4 ] -
Set the exponents equal
Because the bases are identical, the exponents must be equal.
[ 2x+1 = 4 ] -
Solve for (x)
[ 2x = 3 ]
[ x = \frac{3}{2} ]
Answer: (x = \frac{3}{2})
Problem 2 – Logarithmic Equation
Question:
[ \log_{2}(x+3)=5 ]
Solution Steps
-
Rewrite in exponential form
[ x+3 = 2^5 ] -
Compute (2^5)
[ 2^5 = 32 ] -
Isolate (x)
[ x = 32 - 3 = 29 ]
Answer: (x = 29)
Problem 3 – Change of Base Formula
Question:
[ \log_{5}(125) ]
Solution Steps
-
Identify that (125 = 5^3)
[ \log_{5}(5^3) ] -
Apply the power rule
[ \log_{5}(5^3) = 3 \log_{5}(5) ] -
Since (\log_{5}(5)=1)
[ 3 \times 1 = 3 ]
Answer: (3)
Problem 4 – Exponential Growth Model
Question:
A bacterial culture grows according to (N(t)=N_0 e^{kt}).
Given that (N_0=200) cells and after 4 hours the culture has 1600 cells, find the growth constant (k) Took long enough..
Solution Steps
-
Set up the equation
[ 1600 = 200 e^{k \cdot 4} ] -
Divide both sides by 200
[ 8 = e^{4k} ] -
Take natural logarithm
[ \ln 8 = 4k ] -
Solve for (k)
[ k = \frac{\ln 8}{4} \approx \frac{2.07944}{4} \approx 0.51986 ]
Answer: (k \approx 0.52) per hour
Problem 5 – Logarithmic Inequality
Question:
Solve for (x):
[ \log_{3}(x-1) \ge 2 ]
Solution Steps
-
Rewrite in exponential form
[ x-1 \ge 3^2 ] -
Compute (3^2 = 9)
[ x-1 \ge 9 ] -
Add 1 to both sides
[ x \ge 10 ]
Answer: (x \ge 10)
Problem 6 – Graph Interpretation
Question:
A graph of (y = \log_{2}(x)) is given. Identify the x‑intercept Most people skip this — try not to..
Solution
Set (y=0):
[ 0 = \log_{2}(x) \Rightarrow x = 2^0 = 1 ]
So the x‑intercept is at ((1,0)).
Answer: ((1,0))
Problem 7 – Composite Function
Question:
Let (f(x)=2^x) and (g(x)=\log_{2}(x)). Find ((f \circ g)(x)).
Solution
-
Compute (f(g(x)))
[ f(g(x)) = 2^{\log_{2}(x)} ] -
Use the identity (a^{\log_{a}(b)} = b)
[ 2^{\log_{2}(x)} = x ]
Answer: ((f \circ g)(x) = x)
Problem 8 – Exponential Decay
Question:
A radioactive substance has a half‑life of 3 years. If you start with 200 g, how much remains after 9 years?
Solution Steps
-
Use the decay formula
[ N(t) = N_0 \left(\frac{1}{2}\right)^{t/T} ]
where (T=3) years. -
Plug in values
[ N(9) = 200 \left(\frac{1}{2}\right)^{9/3} = 200 \left(\frac{1}{2}\right)^3 ] -
Compute
[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} ]
[ N(9) = 200 \times \frac{1}{8} = 25 \text{ g} ]
Answer: 25 g
Problem 9 – Solving a System with Logarithms
Question:
Solve the system:
[
\begin{cases}
\log_{10}(x) + \log_{10}(y) = 3 \
x - y = 8
\end{cases}
]
Solution Steps
-
Combine logs
[ \log_{10}(xy) = 3 \Rightarrow xy = 10^3 = 1000 ] -
Now we have
[ \begin{cases} xy = 1000 \ x - y = 8 \end{cases} ] -
Express (x) in terms of (y)
[ x = y + 8 ] -
Substitute into product equation
[ (y+8)y = 1000 ]
[ y^2 + 8y - 1000 = 0 ] -
Solve quadratic
Discriminant: (8^2 + 4 \times 1000 = 64 + 4000 = 4064)
[ y = \frac{-8 \pm \sqrt{4064}}{2} ] -
Compute (\sqrt{4064} \approx 63.73)
[ y_1 = \frac{-8 + 63.73}{2} \approx 27.87 \ y_2 = \frac{-8 - 63.73}{2} \approx -35.87 ] -
Reject negative (y) (since (\log_{10}(y)) requires (y>0)).
So (y \approx 27.87) And that's really what it comes down to.. -
Find (x)
[ x = y + 8 \approx 35.87 ]
Answer: (x \approx 35.87,; y \approx 27.87)
Problem 10 – Logarithmic Function Transformation
Question:
Sketch the graph of (y = \log_{2}(x-3) + 1). Identify the asymptote and intercepts That's the part that actually makes a difference..
Solution
- Horizontal shift: (x-3) shifts the basic (\log_{2}(x)) graph right by 3 units.
- Vertical shift: +1 shifts the graph up by 1 unit.
- Vertical asymptote: Set (x-3=0 \Rightarrow x=3).
- x‑intercept: Solve (0 = \log_{2}(x-3) + 1 \Rightarrow \log_{2}(x-3) = -1 \Rightarrow x-3 = 2^{-1} = \frac{1}{2} \Rightarrow x = 3.5).
- y‑intercept: Set (x=0) (not in domain because (x>3)), so no y‑intercept.
Answer:
- Vertical asymptote at (x=3).
- x‑intercept at ((3.5,0)).
- No y‑intercept.
Scientific Explanation – Why These Rules Work
- Exponentiation with a common base: (a^m = a^n \Rightarrow m=n) because exponentiation is a one‑to‑one function for (a>0, a \neq 1).
- Logarithm properties: (\log_a(bc)=\log_a b + \log_a c) and (\log_a(b^k)=k\log_a b) derive from the definition of logarithms as inverse functions of exponentials.
- Growth/decay models: (N(t)=N_0 e^{kt}) comes from solving (\frac{dN}{dt}=kN), a first‑order linear differential equation.
FAQ
| Question | Answer |
|---|---|
| *Can I use any base for logs in these problems? | |
| How do I check if my exponential solution is correct? | Logarithms are undefined for non‑positive arguments in the real number system. * |
| *What if the logarithm argument is negative? * | Substitute back into the original equation; if both sides are equal, it’s correct. |
Conclusion
The problems in Homework 4 reinforce key concepts: rewriting equations with a common base, applying the properties of logarithms, interpreting graphs, and modeling real‑world phenomena with exponential functions. By following the systematic approach outlined above, you can confidently solve similar exercises and deepen your understanding of exponential and logarithmic behavior Most people skip this — try not to. Took long enough..
Continuation of the Conclusion
The mastery of exponential and logarithmic functions extends far beyond textbook exercises. These mathematical tools are foundational in modeling real-world phenomena, from calculating compound interest in finance to predicting population dynamics in ecology. To give you an idea, exponential growth models help scientists forecast the spread of diseases or analyze investment returns, while logarithmic scales are essential in measuring earthquake magnitudes (Richter scale) or sound intensity (decibels). By understanding how to manipulate and interpret these functions, students gain the ability to solve complex problems in diverse fields. The systematic approaches demonstrated in Homework 4—such as isolating variables, applying logarithmic identities, or analyzing graph transformations—equip learners with a versatile toolkit. As technology continues to evolve, computational tools can handle more nuanced calculations, but the conceptual understanding of these principles remains indispensable. Thus, the journey through exponential and logarithmic equations is not just an academic exercise; it is a gateway to applying mathematical reasoning in an increasingly data-driven world.
Final Conclusion
The short version: Homework 4 emphasizes the interplay between algebraic manipulation and graphical interpretation in solving exponential and logarithmic equations. The step-by-step methodologies illustrated—whether solving for variables through exponentiation, leveraging logarithmic properties, or transforming functions for graphing—highlight the logical structure underlying these mathematical concepts. Beyond academic success, these skills build critical thinking and problem-solving abilities applicable to scientific, technological, and everyday challenges. As learners progress, they will encounter more advanced topics, such as logarithmic differentiation or complex exponential systems, but the principles established here will remain a cornerstone. By embracing both the theory and practical applications, students can confidently figure out the vast landscape of mathematics and its myriad applications.
This continuation expands the conclusion to underscore real-world relevance and future learning, ensuring a cohesive and comprehensive ending to the article.
