Exponential & Logarithmic Functions – definitions, properties, graphs, solving techniques

1. Core Definitions

1.1 Domain & Range

• For $y=a^x$: Domain $\mathbb{R}$; Range $(0,\infty )$.
• For $y={\log }_{a}x$: Domain $(0,\infty )$; Range $\mathbb{R}$.

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2. Graphical Characteristics

• Both curves pass the reflection test across the line $y=x$ (they are inverses).
• Exponential curve never touches the x‑axis (horizontal asymptote $y=0$).
• Logarithmic curve never touches the y‑axis (vertical asymptote $x=0$).

2.1 Transformations

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3. Fundamental Properties

1. Product & Quotient
   $$a^m{a}^n=a^{m+n},\frac{a^m}{a^n}=a^{m-n}$$

2. Power of a Power
   $$(a^m{)}^n=a^{mn}$$

3. Logarithm Laws
   $${\log }_a(xy)={\log }_{a}x+{\log }_{a}y,{\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y,{\log }_a(x^k)=k{\log }_{a}x$$

4. Change of Base
   $${\log }_{a}x=\frac{\ln x}{\ln a}$$

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4. Solving Exponential Equations

Example 1 – Compound Interest (Worked Example)

A bank offers 12 % annual interest compounded yearly. Find the amount after 5 years for an initial deposit of NPR 50,000.

$$A=P(1+r{)}^n=50,000(1+0.12{)}^5$$

Compute step‑by‑step:

Answer: NPR 88,115 (≈).

Example 2 – Solving an Exponential Equation

Solve $2^{3x-1}=5$.

1. Take natural log both sides: $\ln 2^{3x-1}=\ln 5$.
2. Use power rule: $(3x-1)\ln 2=\ln 5$.
3. Isolate $x$:
   $$3x-1=\frac{\ln 5}{\ln 2}\Rightarrow x=\frac{1}{3}\left(\frac{\ln 5}{\ln 2}+1\right)$$
4. Numerical value: $\ln 5\approx 1.6094,\ln 2\approx 0.6931$ → $\frac{\ln 5}{\ln 2}\approx 2.3219$.
   $$x=\frac{1}{3}(2.3219+1)=\frac{3.3219}{3}=1.1073$$

Solution: $x\approx 1.11$.

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5. Solving Logarithmic Equations

Example 3 – Logarithmic Identity

Show that for $-1<x<1$,

$$f\left(\frac{2ab}{1+a^2{b}^2}\right)=2f(ab),f(x)=\log \frac{1-x}{1+x}.$$

Proof Sketch

1. Write $f\left(\frac{2ab}{1+a^2{b}^2}\right)=\log \frac{1-\frac{2ab}{1+a^2{b}^2}}{1+\frac{2ab}{1+a^2{b}^2}}$.

2. Simplify numerator and denominator by common denominator $1+a^2{b}^2$:

   $$\frac{(1+a^2{b}^2)-2ab}{(1+a^2{b}^2)+2ab}=\frac{(1-ab{)}^2}{(1+ab{)}^2}.$$

3. Apply log property $\log \frac{u^2}{v^2}=2\log \frac{u}{v}$:

   $$=2\log \frac{1-ab}{1+ab}=2f(ab).$$

Thus the identity holds.

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6. Domain & Range of Rational‑Log Functions

Example 4 – Find Domain & Range

$f(x)=\frac{2x+1}{3-x}$.

Domain: Denominator ≠ 0 → $3-x\ne 0$ → $x\ne 3$. Hence $\text{Dom}=\mathbb{R}\setminus \{3\}$.

Range: Set $y=\frac{2x+1}{3-x}$. Solve for $x$:

$$y(3-x)=2x+1\Rightarrow 3y-yx=2x+1\Rightarrow (2+y)x=3y-1.$$

If $y\ne -2$, $x=\frac{3y-1}{2+y}$ is real, so all real $y$ except the value that makes denominator zero, i.e. $y\ne -2$.

Thus $\text{Range}=\mathbb{R}\setminus \{-2\}$.

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7. Comparison Table

| Feature | Exponential \(a^{x}\) | Logarithmic \(\log_{a}x\) |
|---------|-----------------------|---------------------------|
| Inverse | Yes (log) | Yes (exp) |
| Domain | \(\mathbb{R}\) | \((0,\infty)\) |
| Range | \((0,\infty)\) | \(\mathbb{R}\) |
| Asymptote | Horizontal \(y=0\) | Vertical \(x=0\) |
| Growth | Rapid (if \(a>1\)) | Slow (unbounded) |
| Common Base in Calculus | \(e\) (natural) | \(e\) (natural log) |

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8. Applications

8.1 Real‑World Section

eSewa & Khalti – Transaction Volume
Both platforms experience exponential growth in daily transactions during festive seasons. If the number of transactions $T$ follows $T=T_0{e}^{kt}$ with $k=0.15$ per month, a base of 10,000 transactions in January becomes about $10,000e^{0.15\times 3}\approx 15,000$ by April.

Ncell – Signal Strength (dB)
Signal strength in decibels is a logarithmic measure:
$$\text{dB}=10{\log }_{10}\left(\frac{P_{\text{received}}}{P_{\text{reference}}}\right).$$
If a tower receives $P_{\text{received}}=0.001$ W while the reference is $1$ W, the dB value is $10{\log }_{10}(0.001)=-30$ dB.

Daraz – Order Queue Processing
The average waiting time $W$ for orders in a queue that doubles every hour (e.g., during flash sales) follows an exponential model $W=W_0{e}^{\lambda t}$. With $\lambda =0.5$ h${}^{-1}$ and $W_0=2$ min, after 3 h the wait becomes $2e^{1.5}\approx 9$ min.

These examples illustrate how the same mathematical ideas appear in everyday digital services familiar to Nepali students.

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9. Worked Example Tied to a Real Situation

Problem: A Nepali bank offers a loan with an annual interest rate of 9 % compounded monthly. If a student borrows NPR 200,000, what will be the amount owed after 2 years?

Solution Steps

1. Convert annual rate to monthly: $r=0.09/12=0.0075$.
2. Number of periods: $n=2\times 12=24$.
3. Apply compound formula:
   $$A=P(1+r{)}^n=200,000(1+0.0075{)}^{24}.$$
4. Compute $(1.0075{)}^{24}$ ≈ 1.197 (using a calculator).
5. Multiply: $200,000\times 1.197=239,400$.

Result: After 2 years the student owes NPR 239,400.

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10. Common Mistakes & How to Avoid Them

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11. Quick Reference Formulas

• Exponential growth/decay: $N(t)=N_0{e}^{kt}$ ( $k>0$ growth, $k<0$ decay).
• Logarithmic conversion: ${\log }_{a}b=\frac{\ln b}{\ln a}$.
• Derivative: $\frac{d}{dx}{a}^x=a^x\ln a$; $\frac{d}{dx}\ln x=\frac{1}{x}$.
• Integral: $\int a^{x}dx=\frac{a^x}{\ln a}+C$; $\int \frac{dx}{x}=\ln |x|+C$.

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12. Exam tip

• Identify the function type first. If the problem asks for domain/range, write down the generic domain/range of exponential vs. logarithmic before substituting the specific expression.
• When solving equations, always take logs (or exponentials) on both sides and simplify using the power rule; avoid cross‑multiplying logs directly.
• For transformation questions, remember the order: start with the innermost horizontal shift, then stretch/compress, then vertical shift. Sketch the parent graph and apply each step sequentially.
• Mark asymptotes early; they earn partial credit even if the full graph is not perfect.
• Time‑saving trick: Use the natural base $e$ whenever possible; calculators have an “$e^x$” key and “ln” key, which reduces arithmetic errors.

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