Exponential & Logarithmic Functions: Growth, Decay, and Applications

TAKEAWAYS:

• Exponential functions model compound growth/decay (e.g., investments, population, depreciation) via $f(x)=a\cdot b^x$.
• Logarithms reverse exponentials and solve for time/rate (e.g., half-life, interest rates) using ${\log }_b(a)=c$.
• Graphs of exponentials are asymptotic (never touch axes), while logarithms have vertical asymptotes at $x=0$.
• Change of base formula (${\log }_b(a)=\frac{{\log }_k(a)}{{\log }_k(b)}$) simplifies calculations with any base.
• Applications include loan interest (banks), bacterial growth (medicine), and decibel scales (sound engineering).
• Exam focus: Solve equations, sketch graphs, and interpret real-world scenarios (e.g., Ncell’s data plans, Daraz’s discount rates).

1. Exponential Functions: The Engine of Growth and Decay

Exponential functions describe rapid change—doubling, halving, or multiplying by a fixed factor over equal intervals. Their general form is: $$f(x)=a\cdot b^x$$ where:

• $a$: initial value (at $x=0$),
• $b$: growth factor ($b>1$) or decay factor ($0<b<1$),
• $x$: exponent (time, steps, etc.).

Key Properties

• Growth ($b>1$): Function increases steeply (e.g., investments, bacteria).
• Decay ($0<b<1$): Function decreases toward zero (e.g., radioactive decay, depreciation).
• Asymptote: Horizontal line $y=0$ (the function never touches it).
• Domain: All real numbers ($x\in \mathbb{R}$).
• Range: $y>0$ (always positive).

Worked Example 1: Compound Interest (Nepal’s NMB Bank)

Scenario: You deposit Rs. 50,000 in a bank offering 8% annual interest compounded yearly. How much will you have after 5 years? Solution: Use the compound interest formula (a special exponential function): $$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$ Here:

• $P=50,000$ (principal),
• $r=0.08$ (8% rate),
• $n=1$ (compounded yearly),
• $t=5$ years.

Plugging in: $$A=50,000{\left(1+0.08\right)}^5=50,000\times (1.08{)}^5$$ $$A=50,000\times 1.4693\approx \text{Rs. }73,465$$

Real-World Tie-In: Nepal’s NMB Bank uses this exact formula for fixed-deposit calculations. If you see a question about "interest compounded annually," this is the formula to apply.

2. Logarithmic Functions: The Inverse of Exponentials

Logarithms undo exponentials. If $y=b^x$, then $x={\log }_b(y)$. General Form: $$f(x)={\log }_b(x)$$ where:

• $b$: base ($b>0,b\ne 1$),
• $x$: argument (must be $x>0$).

Key Properties

Worked Example 2: Half-Life of a Drug (Pharmaceuticals)

Scenario: A drug’s concentration halves every 6 hours. If the initial dose is 100 mg, how long until 12.5 mg remains? Solution: This is exponential decay with half-life $t_{1/2}=6$ hours. The decay formula is: $$C(t)=C_0{\left(\frac{1}{2}\right)}^{t/t_{1/2}}$$ We want $C(t)=12.5$: $$12.5=100{\left(\frac{1}{2}\right)}^{t/6}$$ Divide both sides by 100: $$0.125={\left(\frac{1}{2}\right)}^{t/6}$$ Recognize $0.125=\frac{1}{8}={\left(\frac{1}{2}\right)}^3$: $${\left(\frac{1}{2}\right)}^3={\left(\frac{1}{2}\right)}^{t/6}$$ Equate exponents: $$3=\frac{t}{6}$$ $$t=18\text{ hours}$$

Using Logarithms: Alternatively, take the log (base 2) of both sides: $${\log }_2(0.125)=\frac{t}{6}{\log }_2\left(\frac{1}{2}\right)$$ $$-3=\frac{t}{6}\times (-1)$$ $$t=18\text{ hours}$$

Real-World Tie-In: Hospitals use logarithmic calculations to determine drug dosage timing. For example, Paracetamol’s half-life (~4 hours) helps doctors schedule repeated doses.

3. Change of Base Formula: The Universal Logarithm Tool

Most calculators only compute natural logs (ln, base $e$) or common logs (log, base 10). The change of base formula lets you convert any base: $${\log }_b(a)=\frac{{\log }_k(a)}{{\log }_k(b)}$$ where $k$ is any positive number ($k\ne 1$).

Worked Example 3: pH Scale (Chemistry)

Scenario: The pH of a solution is given by $\text{pH}=-{\log }_{10}[{\text{H}}^{+}]$. If $[{\text{H}}^{+}]=0.001$ M, what is the pH? Solution: $$\text{pH}=-{\log }_{10}(0.001)$$ Since $0.001=10^{-3}$: $$\text{pH}=-(-3)=3$$

But what if the base isn’t 10? Use the change of base formula: $${\log }_b(a)=\frac{\ln (a)}{\ln (b)}$$

Example: Convert ${\log }_2(8)$ to natural logs. $${\log }_2(8)=\frac{\ln (8)}{\ln (2)}=\frac{2.079}{0.693}\approx 3$$ (Which matches $2^3=8$.)

Real-World Tie-In:

• NTC’s signal strength is measured in decibels (dB), which uses logarithms: $$\text{dB}=10{\log }_{10}\left(\frac{P_1}{P_0}\right)$$
• YouTube’s algorithm uses logarithmic scaling to rank video views (a linear increase in views doesn’t feel as impactful as exponential growth).

4. Solving Exponential and Logarithmic Equations

Type 1: Exponential Equations

Example: Solve $3^{x+1}=27$. Steps:

1. Express 27 as a power of 3: $27=3^3$.
2. Rewrite the equation: $3^{x+1}=3^3$.
3. Since bases are equal, set exponents equal: $x+1=3$.
4. Solve for $x$: $x=2$.

Type 2: Logarithmic Equations

Example: Solve ${\log }_4(x)+{\log }_4(2)=2$. Steps:

1. Combine logs using ${\log }_b(m)+{\log }_b(n)={\log }_b(mn)$: $${\log }_4(2x)=2$$
2. Rewrite in exponential form: $4^2=2x$.
3. Solve for $x$: $16=2x$ → $x=8$.

Type 3: Mixed Equations (Use Logs)

Example: Solve $5^x=12$. Steps:

1. Take the natural log of both sides: $\ln (5^x)=\ln (12)$.
2. Use the power rule $\ln (a^b)=b\ln (a)$: $$x\ln (5)=\ln (12)$$
3. Solve for $x$: $$x=\frac{\ln (12)}{\ln (5)}\approx 1.544$$

flowchart TD
    A["Exponential Equation: 3^(x+1) = 27"] --> B["Rewrite 27 as 3^3"]
    B --> C["Set exponents equal: x+1=3"]
    C --> D["Solve: x=2"]
    E["Logarithmic Equation: log4(x) + log4(2) = 2"] --> F["Combine logs: log4(2x)=2"]
    F --> G["Exponential form: 4^2=2x"]
    G --> H["Solve: x=8"]
    I["Mixed Equation: 5^x=12"] --> J["Take ln: x*ln(5)=ln(12)"]
    J --> K["Solve: x=ln(12)/ln(5)"]

Real-World Tie-In:

• Pathao’s surge pricing uses exponential functions to adjust fares during peak hours.
• Daraz’s discount rates often follow logarithmic scaling (e.g., "50% off" feels more impactful than "10% off").

5. Applications in Business and Finance

A. Compound Interest vs. Simple Interest

Worked Example 4: Ncell’s Data Plan Scenario: Ncell offers a plan where data doubles every month (exponential growth). If you start with 1 GB, how much data will you have after 4 months? Solution: This is exponential growth with $b=2$: $$\text{Data}=1\times 2^4=16\text{ GB}$$

B. Depreciation of Assets (Business Accounting)

Scenario: A machine costs Rs. 200,000 and depreciates at 10% annually. What’s its value after 3 years? Solution: Use exponential decay ($b=0.9$): $$V(t)=200,000\times (0.9{)}^3$$ $$V(3)=200,000\times 0.729=\text{Rs. }145,800$$

C. Logarithms in Finance: Rule of 72

The Rule of 72 estimates how long it takes for an investment to double: \text{Years to double} = \frac{72}{\text{Interest Rate (%)}} Example: At 6% interest, doubling time = $\frac{72}{6}=12$ years.

mindmap
  root((Business Applications))
    Compound Interest
      Banks: Fixed Deposits
      Investments: Stocks, Bonds
    Depreciation
      Accounting: Asset Value
      Tax Calculations
    Logarithms
      Rule of 72: Doubling Time
      pH Scale: Chemistry
    Technology
      Algorithms: YouTube, Google
      Signal Strength: NTC, Ncell

6. Graphs of Exponential and Logarithmic Functions

Exponential Graphs ($y=b^x$)

• Growth ($b>1$): Starts near 0, rises steeply.
• Decay ($0<b<1$): Starts high, flattens toward 0.
• Key Points: Always passes through $(0,1)$ if $a=1$.

Logarithmic Graphs ($y={\log }_b(x)$)

• Increasing ($b>1$): Passes through $(1,0)$, rises slowly.
• Decreasing ($0<b<1$): Passes through $(1,0)$, falls slowly.
• Key Points: Never crosses $x=0$ (vertical asymptote).

## In the Real World

1. Khalti’s Transaction Fees:

   • Khalti charges a flat fee + percentage of transactions. The percentage acts like an exponential decay on your balance if you frequently transfer small amounts.
   • Example: If you transfer Rs. 1,000 and Khalti takes 2%, your effective balance grows exponentially slower than a lump sum.

2. Daraz’s Discount Pricing:

   • Daraz’s "Buy 1 Get 1 Free" deals can be modeled using exponential functions for bulk discounts. For example, buying 4 items at a 50% discount each is better than buying 2 at 25% off (logarithmic scaling of perceived value).

3. Nepal Stock Exchange (NEPSE) Index:

   • The NEPSE index uses logarithmic scaling to represent percentage changes in stock prices. A 10% drop followed by a 10% rise doesn’t return to the original value (due to exponential decay in percentages).

## Exam Tip

1. Graphs Are Worth Marks:

   • Always label axes, asymptotes, and key points (e.g., $(0,1)$ for $y=b^x$).
   • Common Mistake: Forgetting the asymptote or mislabeling the base.

2. Equation Solving:

   • For $a^{f(x)}=b$, take the log of both sides: $f(x)\log (a)=\log (b)$.
   • Shortcut: If the equation has the same base, set exponents equal directly.

3. Real-World Scenarios:

   • Banks: Use compound interest formulas.
   • Technology: Logarithms appear in signal strength (dB), algorithms (YouTube rankings).
   • Science: Half-life (medicine), pH scale (chemistry).

4. Change of Base Formula:

   • Memorize: ${\log }_b(a)=\frac{\ln (a)}{\ln (b)}$.
   • Use it when your calculator only has $\ln$ or ${\log }_{10}$.

5. Units and Context:

   • Always include units (e.g., "Rs. X after Y years").
   • Example: "The population grows exponentially at 5% per year" → $P(t)=P_0(1.05{)}^t$.

Final Checklist Before Submitting:

• Drew graphs for exponential growth/decay and logarithmic functions.
• Solved at least one compound interest and one logarithmic equation.
• Linked examples to real-world scenarios (banks, Ncell, Daraz).
• Used the change of base formula in a worked example.
• Highlighted exam tips for graphing and equation solving.