Day -2: The Laws of Exponents & The Natural Base e and ln(x) & The Genesis of e & Growth, Decay, and Continuous Compounding & Log Properties and Equations & Growth Hierarchy at Infinity

The Laws of Exponents

Before calculus, you must master the 'grammar' of exponents. These rules allow us to rewrite complex expressions into power forms that are easy to differentiate.

Core Theorem

Properties (for

a, b > 0

):
1.

a^x \cdot a^y = a^{x+y}

.
2.

\frac{a^x}{a^y} = a^{x-y}

.
3.

(a^x)^y = a^{xy}

.
4.

a^x \cdot b^x = (ab)^x

.
5.

a^{-x} = \frac{1}{a^x}

.
6.

a^{1/n} = \sqrt[n]{a}

.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Easy

Simplify and write as a single base:

\frac{(e^3)^4 \cdot e^{-2}}{e^5}

.

How to think about it

Key idea: Follow the Order of Operations: Powers first, then multiplication/division.

Full solution

1. Step 1: Power of a Power

(e^3)^4 = e^{3 \cdot 4} = e^{12}

.

2. Step 2: Combine Numerator

e^{12} \cdot e^{-2} = e^{12 + (-2)} = e^{10}

.

3. Step 3: Division Rule

\frac{e^{10}}{e^5} = e^{10-5} = e^5

.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Easy

Solve for

x

:

3^{4x-1} = 27

.

Need a hint?

Rewrite both sides to have the same base.

Show solution

1. Step 1: Match Bases

Notice

27 = 3^3

. So,

3^{4x-1} = 3^3

.

2. Step 2: Equate Exponents

Since bases are equal,

4x - 1 = 3

.

3. Step 3: Solve

4x = 4 \implies x = 1

.

The Natural Base e and ln(x)

The number e (approx 2.718) is the 'natural' base of growth. The natural logarithm $ln(x)$ is its inverse.

Core Theorem

Inverse Properties:
- 1.

e^{ln x} = x

for

x > 0

.
- 2.

ln(e^x) = x

for all

real

x

.
- 3.

y = e^x \iff x = ln y

.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Easy

Solve for

x

:

e^{2x - 3} = 5

.

How to think about it

Key idea: Use $ln$ to 'kill' the $e$ .

Full solution

1. Step 1: Apply Natural Log

Take

ln

of both sides:

ln(e^{2x-3}) = ln(5)

.

2. Step 2: Simplify with Inverse Property

The left side becomes

2x - 3 = ln 5

.

3. Step 3: Solve for x

2x = ln 5 + 3 \implies x = \frac{ln 5 + 3}{2}

.

Step-by-Step SOP

1
Isolate the Exponential
Before taking $ln$ , make sure the $e^{...}$ term is by itself.
2
Check Domain
Always ensure the argument inside a $ln$ is positive.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

Simplify

f(x) = e^{3 ln x}

.

Need a hint?

You cannot cancel

e

and

ln

if there is a number (the 3) in between them. Move the 3 first!

Show solution

1. Step 1: Move the Coefficient

Using log properties,

3 ln x = ln(x^3)

.

2. Step 2: Apply Inverse Property

e^{ln(x^3)} = x^3

.

Common Pitfalls

⚠
Negative Input Trap $ln(x)$ is only defined for $x > 0$ . If you solve an equation and get $x = -2$ , it is an extraneous solution.
⚠
The 'e' is not a VariableRemember $e$ is a constant ( $approx 2.718$ ). Do not treat it like $x$ or $y$ .

The Genesis of e

Beyond just a number, e is defined by the limit of continuous compounding and the geometry of slopes.

Core Theorem

1. Limit definition:

e = \lim_{x \to \infty} (1 + \frac{1}{x})^x

.
2. Calculus property: The slope of

y=e^x

at

x=0

is

1

.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

If

1000

is invested at

5.5\%

interest compounded continuously, how long until it doubles?

How to think about it

Key idea: Use $A = Pe^{rt}$ and solve for $t$ when $A = 2P$ .

Full solution

1. Setup Equation

2000 = 1000e^{0.055t} \implies 2 = e^{0.055t}

.

2. Apply ln

\ln 2 = 0.055t

.

3. Solve

t = \frac{ln 2}{0.055} \approx 12.6

years.

Growth, Decay, and Continuous Compounding

In the real world, growth doesn't always happen in chunks. Continuous compounding describes systems (like interest or bacteria) that grow at every single instant.

Core Theorem

1. Discrete Compounding:

A = P(1 + \frac{r}{n})^{nt}

2. Continuous Growth/Decay:

A = Pe^{rt}

(or

y = y_0 e^{kt}

)

Notation Key:
*

A

(or

y

): Final amount / Future value.
*

P

(or

y_0

): Principal / Initial amount at

t=0

.
*

r

(or

k

): Annual rate / Growth constant (use decimals:

5\% = 0.05

).
*

n

: Number of compounding periods per year.
*

t

: Time elapsed (usually in years).

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Easy

An investment firm uses continuous compounding. If you invest

100

at an annual rate of

5.5\%

, what is the total amount after 4 years?

How to think about it

Key idea: Identify your variables: $P=100, r=0.055, t=4$ . Use $Ae^{rt}$ .

Full solution

1. Step 1: Set up the formula

A = 100 \cdot e^{0.055 \cdot 4}

2. Step 2: Simplify exponent

A = 100 \cdot e^{0.22}

3. Step 3: Final Calculation

A \approx 100 \cdot 1.246 = 124.6

. The total is approx

124.6

dollars.

Step-by-Step SOP

1
Identify the 'n'
If the problem says 'compounded monthly', use the power $12t$ . If it says 'continuously', use $e^{rt}$ .
2
Finding k
In many calculus problems, you must first solve for $k$ using a known data point before answering the main question.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

Carbon-14 has a decay constant

k = 1.2 \cdot 10^{-4}

. If an initial amount

A

is present, how much remains after 866 years?

Need a hint?

Decay is just growth with a negative

k

. Use

y = A e^{-kt}

.

Show solution

1. Step 1: Setup Model

y = A \cdot e^{-(1.2 \cdot 10^{-4}) \cdot 866}

2. Step 2: Calculate Exponent

-(0.00012) \cdot 866 \approx -0.10392

3. Step 3: Result

y = A \cdot e^{-0.10392} \approx 0.901A

. About

90\%

remains.

Common Pitfalls

⚠
The Percentage MistakeAlways convert percentages to decimals. $5.5\%$ must be $0.055$ , not $5.5$ .
⚠
Growth vs. DecayA common error is forgetting the negative sign in decay problems. If the amount isn't shrinking, check your $k$ !

Log Properties and Equations

Logarithms turn multiplication into addition and powers into coefficients. Mastering these allows us to solve complex exponential equations and perform 'Logarithmic Differentiation'.

Core Theorem

Laws of Logs (for

a, b, x, y > 0

):
1. Product:

\ln(xy) = \ln x + \ln y

2. Quotient:

\ln(x/y) = \ln x - \ln y

3. Power:

\ln(x^p) = p \ln x

4. Base Change:

\log_a x = \frac{\ln x}{\ln a}

5. Special Values:

\ln e = 1

,

\ln 1 = 0

6. Identity:

a^x = e^{x \ln a}

(Crucial for Calculus!)

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

Expand completely:

\ln\left(\frac{x^2 \sqrt{y}}{z^3}\right)

.

How to think about it

Key idea: Numerator terms become positive logs; denominator terms become negative logs.

Full solution

1. Step 1: Split Top/Bottom

\ln(x^2 \sqrt{y}) - ln(z^3)

2. Step 2: Split Multiplication

\ln(x^2) + \ln(y^{1/2}) - \ln(z^3)

3. Step 3: Power Rule

2 \ln x + \frac{1}{2} \ln y - 3 \ln z

Step-by-Step SOP

1
Identify Bases
If you see different bases in one equation, use the Change of Base formula to convert everything to $\ln$ .

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

Solve the inequality:

\ln(x^2 - 2x - 2) \leq 0

.

Need a hint?

Step 1: Inside must be

>0

. Step 2: Use

e

to cancel

\ln

.

Show solution

1. Step 1: Domain Check

Set

x^2 - 2x - 2 > 0

. Using quadratic formula,

x < 1-\sqrt{3}

or

x > 1+\sqrt{3}

.

2. Step 2: Solve Inequality

Apply

e^{...}

to both sides:

x^2 - 2x - 2 \leq e^0 \implies x^2 - 2x - 2 \leq 1

.

3. Step 3: Final Range

x^2 - 2x - 3 \leq 0 \implies (x-3)(x+1) \leq 0 \implies -1 \leq x \leq 3

. Combined with Domain,

x \in [-1, 1-\sqrt{3}) \cup (1+\sqrt{3}, 3]

.

Practice 02Medium

Solve for

x

:

3^{\log_3 7} - 4^{\log_4 2} = 5^{(\log_5 x - \log_5 x^2)}

.

Need a hint?

Use the Inverse Identity:

a^{\log_a M} = M

.

Show solution

1. Step 1: Simplify Left Side

7 - 2 = 5

.

2. Step 2: Simplify Right Side

Using log properties,

5^{\log_5(x/x^2)} = 5^{\log_5(1/x)} = 1/x

.

3. Step 3: Solve

5 = 1/x \implies x = 1/5

.

Common Pitfalls

⚠
The Domain GhostAlways find the domain *before* you start moving terms around. Some moves (like power rule) can hide domain restrictions.
⚠
Base Trap $\frac{\ln A}{\ln B}$ is the Change of Base formula, NOT $\ln(A-B)$ .

Growth Hierarchy at Infinity

Understanding which functions 'win' as x to infty is essential for L'Hopital's Rule and Taylor Series.

Core Theorem

The Dominance Hierarchy:

e^x > x^n > \ln x

as

x \to \infty

. Any exponential grows faster than any polynomial, which grows faster than any log.

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

Evaluate

\lim_{x \to \infty} \frac{\ln(x^{100})}{x}

.

How to think about it

Key idea: Don't be scared by $x^{100}$ . Use log properties first.

Full solution

1. Step 1: Simplify

ln(x^{100}) = 100 ln x

.

2. Step 2: Compare Growth

We have

100 ln x

(Log) over

x

(Polynomial). Since Polynomials grow faster...

3. Step 3: Result

The denominator wins. The limit is 0.

Step-by-Step SOP

1
Identify the 'Fastest' Term
Look for the dominant term in the numerator and denominator to determine the limit's behavior.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Hard

Which function will eventually have larger values:

f(x) = 1.0001^x

or

g(x) = x^{10000}

?

Need a hint?

Growth hierarchy doesn't care about the size of the constants.

Show solution

1. Analysis

Even though

1.0001

is a tiny base and

x^{10000}

has a huge power,

1.0001^x

is an **Exponential** and

x^{10000}

is a **Polynomial**. At a large enough

x

, the exponential will ALWAYS win.

Common Pitfalls

⚠
The 'Small Base' DelusionMany students think $1.1^x$ is 'small'. At $x \to \infty$ , it will eventually crush $x^2, x^3,$ or even $x^{100}$ .

Worked Examples

1. Step 1: Power of a Power

2. Step 2: Combine Numerator

3. Step 3: Division Rule

Practice — Your Turn

1. Step 1: Match Bases

2. Step 2: Equate Exponents

3. Step 3: Solve

Worked Examples

1. Step 1: Apply Natural Log

2. Step 2: Simplify with Inverse Property

3. Step 3: Solve for x

Isolate the Exponential

Check Domain

Practice — Your Turn

1. Step 1: Move the Coefficient

2. Step 2: Apply Inverse Property

Worked Examples

1. Setup Equation

2. Apply ln

3. Solve

Worked Examples

1. Step 1: Set up the formula

2. Step 2: Simplify exponent

3. Step 3: Final Calculation

Identify the 'n'

Finding k

Practice — Your Turn

1. Step 1: Setup Model

2. Step 2: Calculate Exponent

3. Step 3: Result

Worked Examples

1. Step 1: Split Top/Bottom

2. Step 2: Split Multiplication

3. Step 3: Power Rule

Identify Bases

Practice — Your Turn

1. Step 1: Domain Check

2. Step 2: Solve Inequality

3. Step 3: Final Range

1. Step 1: Simplify Left Side

2. Step 2: Simplify Right Side

3. Step 3: Solve

Worked Examples

1. Step 1: Simplify

2. Step 2: Compare Growth

3. Step 3: Result

Identify the 'Fastest' Term

Practice — Your Turn

1. Analysis

Gary Chang