-2

Day -2

Prep Exponential Log Mastery

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}

Practice Exercises

Example 01Easy

Simplify and write as a single base:

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

NEED A HINT?

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

SHOW DETAILED EXPLANATION

Step 1: Power of a Power

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

Step 2: Combine Numerator

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

Step 3: Division Rule

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

Example 02Easy

Solve for

x

3^{4x-1} = 27

NEED A HINT?

Rewrite both sides to have the same base.

SHOW DETAILED EXPLANATION

Step 1: Match Bases

Notice

27 = 3^3

. So,

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

Step 2: Equate Exponents

Since bases are equal,

4x - 1 = 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

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 Exercises

Example 01Easy

Solve for

x

e^{2x - 3} = 5

NEED A HINT?

Use

ln

to 'kill' the

e

SHOW DETAILED EXPLANATION

Step 1: Apply Natural Log

Take

ln

of both sides:

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

Step 2: Simplify with Inverse Property

The left side becomes

2x - 3 = ln 5

Step 3: Solve for x

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

Example 02Medium

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 DETAILED EXPLANATION

Step 1: Move the Coefficient

Using log properties,

3 ln x = ln(x^3)

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

x=0

1

Practice Exercises

Example 01Medium

1000

is invested at

5.5\%

interest compounded continuously, how long until it doubles?

NEED A HINT?

Use

A = Pe^{rt}

and solve for

t

when

A = 2P

SHOW DETAILED EXPLANATION

Setup Equation

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

Apply ln

\ln 2 = 0.055t

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).

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 Exercises

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?

NEED A HINT?

Identify your variables:

P=100, r=0.055, t=4

. Use

Ae^{rt}

SHOW DETAILED EXPLANATION

Step 1: Set up the formula

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

Step 2: Simplify exponent

A = 100 \cdot e^{0.22}

Step 3: Final Calculation

A \approx 100 \cdot 1.246 = 124.6

. The total is approx

124.6

dollars.

Example 02Medium

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 DETAILED EXPLANATION

Step 1: Setup Model

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

Step 2: Calculate Exponent

-(0.00012) \cdot 866 \approx -0.10392

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!)

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 Exercises

Example 01Medium

Expand completely:

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

NEED A HINT?

Numerator terms become positive logs; denominator terms become negative logs.

SHOW DETAILED EXPLANATION

Step 1: Split Top/Bottom

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

Step 2: Split Multiplication

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

Step 3: Power Rule

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

Example 02Medium

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 DETAILED EXPLANATION

Step 1: Domain Check

Set

x^2 - 2x - 2 > 0

. Using quadratic formula,

x < 1-\sqrt{3}

x > 1+\sqrt{3}

Step 2: Solve Inequality

Apply

e^{...}

to both sides:

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

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]

Example 03Medium

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 DETAILED EXPLANATION

Step 1: Simplify Left Side

7 - 2 = 5

Step 2: Simplify Right Side

Using log properties,

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

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

x \to \infty

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

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 Exercises

Example 01Medium

Evaluate

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

NEED A HINT?

Don't be scared by

x^{100}

. Use log properties first.

SHOW DETAILED EXPLANATION

Step 1: Simplify

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

Step 2: Compare Growth

We have

100 ln x

(Log) over

x

(Polynomial). Since Polynomials grow faster...

Step 3: Result

The denominator wins. The limit is 0.

Example 02Hard

Which function will eventually have larger values:

f(x) = 1.0001^x

g(x) = x^{10000}

NEED A HINT?

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

SHOW DETAILED EXPLANATION

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 $x^{100}$ .

Practice Exercises

Step 1: Power of a Power

Step 2: Combine Numerator

Step 3: Division Rule

Step 1: Match Bases

Step 2: Equate Exponents

Step 3: Solve

Isolate the Exponential

Check Domain

Practice Exercises

Step 1: Apply Natural Log

Step 2: Simplify with Inverse Property

Step 3: Solve for x

Step 1: Move the Coefficient

Step 2: Apply Inverse Property

Practice Exercises

Setup Equation

Apply ln

Solve

Identify the 'n'

Finding k

Practice Exercises

Step 1: Set up the formula

Step 2: Simplify exponent

Step 3: Final Calculation

Step 1: Setup Model

Step 2: Calculate Exponent

Step 3: Result

Identify Bases

Practice Exercises

Step 1: Split Top/Bottom

Step 2: Split Multiplication

Step 3: Power Rule

Step 1: Domain Check

Step 2: Solve Inequality

Step 3: Final Range

Step 1: Simplify Left Side

Step 2: Simplify Right Side

Step 3: Solve

Identify the 'Fastest' Term

Practice Exercises

Step 1: Simplify

Step 2: Compare Growth

Step 3: Result

Analysis