Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Exponential Growth and Decay: Functions of the form $P(t)=P_0(1+r{)}^t$ or $P(t)=P_0{e}^{kt}$ model quantities that increase or decrease at a constant percentage rate. When $r>0$ (or $k>0$) the function grows; when $r<0$ (or $k<0$) it decays toward zero.

Important Formulas

Below are the key formulas used in this exercise:

Worked Examples

Example 1 — Compound Interest (Solving for Time)

Problem: An investment of Rs. 5,000 grows at 6% per year. How many years until it reaches Rs. 10,000?

Solution: $10000=5000(1.06{)}^t$ $2=(1.06{)}^t$ $\log 2=t\log (1.06)\text{(power rule)}$ $t=\frac{\log 2}{\log 1.06}\approx \frac{0.3010}{0.0253}\approx 11.9\text{ years}$

Example 2 — Continuous Growth (Doubling Time)

Problem: A bacterial culture grows continuously at rate $k=0.2$ per hour. Find the doubling time.

Solution: $2P_0=P_0{e}^{0.2t}$ $2=e^{0.2t}$ $\ln 2=0.2t$ $t=\frac{\ln 2}{0.2}=\frac{0.6931}{0.2}\approx 3.47\text{ hours}$

Example 3 — Logarithmic Model (Solving for Time)

Problem: Memory retention is modelled by $S(t)=80-15\log (t+1)$, where $t$ is weeks. Find when retention drops to 50.

Solution: $50=80-15\log (t+1)$ $15\log (t+1)=30$ $\log (t+1)=2$ $t+1=10^2=100$ $t=99\text{ weeks}$

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Summary

This exercise explores practical applications of Exponential and Logarithmic Functions in modelling real-world phenomena such as population dynamics, financial investments, and memory retention.

Key Learnings:

• Exponential functions are characterised by a constant percentage rate of change, leading to a fixed "doubling time" that is independent of the initial amount.

• Logarithms are the essential tool for isolating a variable that appears as an exponent — always apply $\log$ to both sides and use the power rule.

• Graphically, exponential functions ($a>1$) grow increasingly steeply; logarithmic functions grow at a decreasing rate. Both have asymptotes.

Problem-Solving Strategy:

1. Identify the model type (discrete vs. continuous, growth vs. decay).
2. Identify the initial value $P_0$ and the rate $r$ or $k$.
3. Substitute the known target value and solve for $t$ using logarithms.
4. Check units (years, months, weeks) for consistency.