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Week 4: Exponential and Logarithmic Functions

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Week 4: Exponential and Logarithmic Functions

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Now · 1. Exponential Functions

Course: Jan 2026 - Mathematics I Difficulty: Moderate Focus: Growth, decay, inverse functions, and their powerful properties.

1. Exponential Functions

An Exponential Function is a mathematical function of the form:

f(x) = a^x

Where:

$a$ is a positive real number ( $a > 0$ ) and $a \neq 1$ . $a$ is called the base.
$x$ is any real number. $x$ is called the exponent.

1.1 Behaviours and Graphs

The graph of

y = a^x

passes through

(0, 1)

because

a^0 = 1

. The

x

-axis is a horizontal asymptote.

Exponential Growth ( $a > 1$ ): As $x$ increases, $y$ increases rapidly. The curve goes upwards from left to right. Example: Population growth.
Exponential Decay ( $0 < a < 1$ ): As $x$ increases, $y$ decreases, approaching zero but never touching it. The curve goes downwards from left to right. Example: Radioactive decay.

1.2 The Natural Exponential Function

The natural exponential function is

f(x) = e^x

, where

e

is Euler's number (

e \approx 2.71828\dots

). It represents continuous growth processes and is incredibly prominent in calculus.

1.3 Laws of Exponents

For bases

a, b > 0

and any real exponents

x, y

$a^x \cdot a^y = a^{x+y}$
$\frac{a^x}{a^y} = a^{x-y}$
$(a^x)^y = a^{xy}$
$(ab)^x = a^x b^x$
$a^{-x} = \frac{1}{a^x}$

2. Logarithmic Functions

The Logarithmic Function is the inverse operation to exponentiation. It answers the question: "To what power must we raise the base

a

to get the number

x

y = \log_a(x) \iff a^y = x

Where:

Base $a > 0$ and $a \neq 1$ .
Argument $x > 0$ . (We cannot take the logarithm of zero or a negative number).

2.1 Graph of a Logarithmic Function

Because

y = \log_a(x)

is the inverse of

y = a^x

, its graph is the reflection of the exponential graph across the line

y = x

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
The $y$ -axis is a vertical asymptote.
Passes through $(1, 0)$ because $\log_a(1) = 0$ .

2.2 Common Logarithms

Common Logarithm: Base 10. Usually written as $\log(x)$ or $\log_{10}(x)$ .
Natural Logarithm: Base $e$ . Written as $\ln(x)$ . Thus, $\ln(x) = y \iff e^y = x$ .

3. Properties of Logarithms

Since logarithms are exponents, they follow corresponding rules: Let

M, N, a

be positive real numbers (

a \neq 1

), and

p

be any real number.

Product Rule: $\log_a(MN) = \log_a(M) + \log_a(N)$
Quotient Rule: $\log_a(\frac{M}{N}) = \log_a(M) - \log_a(N)$
Power Rule: $\log_a(M^p) = p \cdot \log_a(M)$
Change of Base Formula: $\log_a(M) = \frac{\log_b(M)}{\log_b(a)}$ (often used to convert to base 10 or base $e$ for calculation).

Common pitfalls:

$\log_a(M + N) \neq \log_a(M) + \log_a(N)$ . (Logarithm doesn't distribute over addition).
$\log_a(M) \cdot \log_a(N) \neq \log_a(MN)$ .

4. Exponential and Logarithmic Equations

4.1 Solving Exponential Equations

Same Base Method: If you can express both sides with the same base, equate the exponents. If $a^u = a^v$ , then $u = v$ .
Taking Logarithms: If bases differ, take the logarithm (usually $\ln$ or $\log$ ) of both sides and use the power rule to bring the variable exponent down. $2^x = 7 \implies \ln(2^x) = \ln(7) \implies x \ln(2) = \ln(7) \implies x = \frac{\ln(7)}{\ln(2)}$

4.2 Solving Logarithmic Equations

Condense: Use log properties to combine multiple logs into a single logarithm on each side.
Exponentiate: Convert the equation from logarithmic form to exponential form. $\log_a(x) = y \implies x = a^y$ .
Check for Extraneous Solutions: ALWAYS plug your potential solutions back into the original equation to ensure the arguments of all logarithms are strictly positive. Solutions that produce $\log(\text{negative})$ or $\log(0)$ must be rejected.

📘 Textbook Notes: Week 4 Notes
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