LOGARITHMS & EXPONENTIALS

WHY THIS MATTERS

Nature loves to grow and decay exponentially. Radioactive nuclei decay, capacitors charge and discharge, population grows — all following $e^x$ or $e^{-x}$ curves. Logarithms are the mathematical tool to undo exponentials, turning multiplicative problems into additive ones. This part is essential for Thermodynamics, Nuclear Physics, and Electric Circuits.

§1. Definition of Logarithm

Logarithm is the inverse of exponentiation. Formally:

CORE DEFINITION

$$\log_b(x) = y \quad \iff \quad b^y = x$$

Read as: "The logarithm of $x$ to the base $b$ equals $y$" means $b$ raised to the power $y$ gives $x$.

Key constraint: $b > 0$, $b \neq 1$, and $x > 0$ (you cannot take the log of a negative number or zero!)

Example 1

$\log_2(8) = 3$ because $2^3 = 8$

Example 2

$\log_{10}(1000) = 3$ because $10^3 = 1000$

Example 3

$\log_5(1) = 0$ because $5^0 = 1$

§2. Types of Logarithms

Common Logarithm ($\log_{10}$)

Written as $\log x$ (without base, base 10 is implied)

Used in: decibel scale (Sound), pH scale, Richter scale

Natural Logarithm ($\ln$)

Written as $\ln x = \log_e x$ where $e \approx 2.71828\dots$

Used in: Radioactive decay, RC/RL circuits, Thermodynamics

⭐ CONVERSION FORMULA

$$\ln x = 2.303 \times \log_{10} x$$

This is used constantly when switching between natural log (physics theory) and common log (log tables).

§3. Laws / Properties of Logarithms

These 4 laws are the entire toolkit of logarithmic manipulation. Memorize all 4.

Law 1 — Product Rule

$\log_b(mn) = \log_b m + \log_b n$

Multiplication becomes Addition

Law 2 — Quotient Rule

$\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n$

Division becomes Subtraction

Law 3 — Power Rule

$\log_b(m^n) = n \cdot \log_b m$

Exponent comes down as multiplier

Law 4 — Change of Base

$\log_b m = \frac{\log_c m}{\log_c b}$

Convert any base using this

Special Values — Must Know!

$\log_b 1 = 0$ (always)

$\log_b b = 1$ (always)

$b^{\log_b x} = x$ (inverse)

$\log_b(b^x) = x$ (inverse)

Solved Example 1: Expand $\log\left(\frac{x^3 \sqrt{y}}{z^2}\right)$

$= \log(x^3) + \log(\sqrt{y}) - \log(z^2)$
$= 3\log x + \frac{1}{2}\log y - 2\log z$

Solved Example 2: Solve $\log_2(x) + \log_2(x-2) = 3$

Using product rule: $\log_2[x(x-2)] = 3$
$x(x-2) = 2^3 = 8 \implies x^2 - 2x - 8 = 0$
$(x-4)(x+2) = 0 \implies x = 4 \text{ or } x = -2$
Since $\log$ is undefined for negative values, $\mathbf{x = 4}$.

§4. The Number $e$ & Exponential Functions

Graph: Exponential growth $e^x$, Decay $e^{-x}$, and Natural Log $\ln x$

Euler's number $e \approx 2.71828\dots$ is irrational and transcendental. It can be defined as:

$$e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \dots$$

The Exponential Function $y = e^x$

KEY PROPERTIES OF $e^x$

Domain: all real numbers. Range: $(0, \infty)$ — always positive!
$e^0 = 1$, $e^1 = e \approx 2.718$
Its derivative is itself: $\frac{d}{dx}(e^x) = e^x$ — unique in all of mathematics!
$e^x \cdot e^y = e^{x+y}$ (law of indices)
$e^x$ never equals 0, even as $x \to -\infty$ it approaches 0 asymptotically.

⚛️ Physics Use — Radioactive Decay:
$N(t) = N_0 \cdot e^{-\lambda t}$
where $N_0$ is initial nuclei count and $\lambda$ is the decay constant. Taking $\ln$ on both sides:
$\ln N = \ln N_0 - \lambda t$
This is a straight-line equation with slope $-\lambda$. A log-linear graph of decay is always a straight line!

Decay vs Growth Curves

Growth: $y = e^x$ or $y = e^{+kt}$

Starts from near zero, rises steeply.
Examples: Bacterial growth, capacitor charging current (initially).

Decay: $y = e^{-x}$ or $y = e^{-kt}$

Starts high, falls toward zero (never reaching it).
Examples: Radioactive decay, capacitor discharge, damped oscillations.

§5. Antilogarithm (Antilog)

The antilogarithm is simply the inverse of the logarithm operation. If $\log_{10}(x) = y$, then $x = \text{antilog}(y) = 10^y$.

Example: Find antilog of 2.3010

$x = 10^{2.3010} = 10^2 \times 10^{0.3010} = 100 \times 2 = \mathbf{200}$
(Since $\log_{10}(2) = 0.3010$)

Common Log Values to Memorize

$x$	$\log_{10} x$	$x$	$\log_{10} x$
1	0	6	0.778
2	0.301	7	0.845
3	0.477	8	0.903
4	0.602	9	0.954
5	0.699	10	1.000

Practice Questions

DRILL 1 — BASIC EVALUATION

1. Evaluate $\log_4 64$
2. Evaluate $\log_{10} 0.001$
3. Evaluate $\log_3 \frac{1}{27}$
4. Find $x$ if $\log_x 8 = 3$
5. Find $x$ if $\log_2 x = -4$

DRILL 2 — LAWS OF LOGARITHMS

6. Expand: $\log\left(\frac{a^2 b^3}{c}\right)$
7. Simplify: $2\ln 3 + \ln 5 - \ln 45$
8. Express as single log: $3\log x - \log y + 2\log z$
9. Solve: $\log(3x-1) = 2$
10. Solve: $\log(x+3) + \log(x-3) = \log 7$

DRILL 3 — EXPONENTIALS & PHYSICS

11. Simplify: $e^{3\ln 2}$
12. Solve: $e^{2x} = 7$ (leave answer in log form)
13. The half-life of Carbon-14 is 5730 years. Using $N = N_0 e^{-\lambda t}$, find $\lambda$.
14. A capacitor charges as $q = C\varepsilon(1 - e^{-t/RC})$. At what time $t$ does $q = \frac{1}{2}C\varepsilon$?
15. If $\log 2 = 0.301$ and $\log 3 = 0.477$, find $\log 12$ without a calculator.

DRILL 4 — ADVANCED

16. Prove: $\log_a b \times \log_b a = 1$
17. The sound intensity level in decibels is $\beta = 10\log\left(\frac{I}{I_0}\right)$. If $I = 100 I_0$, find $\beta$.
18. Solve: $2^x = 3^{x-1}$ (use logs)
19. In radioactive decay $N = N_0 e^{-\lambda t}$, if after 10 years 75% has decayed, find $\lambda$.
20. Convert $\ln 50$ to $\log_{10}$ form and find a numerical value (given $\log 2 = 0.301$).

Answer Key

Q	Answer & Method
1	$\log_4 64 = 3$ since $4^3 = 64$
2	$\log_{10} 0.001 = \log_{10} 10^{-3} = \mathbf{-3}$
3	$\log_3 3^{-3} = \mathbf{-3}$
4	$\log_x 8 = 3 \implies x^3=8 \implies x = \mathbf{2}$
5	$x = 2^{-4} = \mathbf{1/16}$
6	$2\log a + 3\log b - \log c$
7	$\ln(9 \times 5) - \ln 45 = \ln 45 - \ln 45 = \mathbf{0}$
8	$\log\left(\frac{x^3 z^2}{y}\right)$
9	$3x - 1 = 100 \implies x = \mathbf{101/3 \approx 33.67}$
10	$\log(x^2-9) = \log 7 \implies x^2 = 16 \implies x = \mathbf{4}$ (rejecting $-4$)
11	$e^{3\ln 2} = e^{\ln 8} = \mathbf{8}$
12	$2x = \ln 7 \implies x = \frac{\ln 7}{2} \approx \mathbf{0.973}$
13	At $t_{1/2}$: $N_0/2 = N_0 e^{-\lambda \cdot 5730} \implies \lambda = \frac{\ln 2}{5730} \approx \mathbf{1.21 \times 10^{-4}\text{ yr}^{-1}}$
14	$\frac{1}{2} = 1 - e^{-t/RC} \implies e^{-t/RC} = 0.5 \implies t = \mathbf{RC\ln 2}$
15	$\log 12 = \log(4 \times 3) = 2(0.301) + 0.477 = \mathbf{1.079}$
16	$\frac{\log b}{\log a} \times \frac{\log a}{\log b} = \mathbf{1}$. $\checkmark$
17	$\beta = 10\log(100) = 10 \times 2 = \mathbf{20\text{ dB}}$
18	$x\log 2 = (x-1)\log 3 \implies x = \frac{\log 3}{\log 3 - \log 2} = \frac{0.477}{0.176} \approx \mathbf{2.71}$
19	25% remains: $0.25 = e^{-10\lambda} \implies \lambda = \frac{\ln 4}{10} \approx \mathbf{0.1386\text{ yr}^{-1}}$
20	$\ln 50 = 2.303 \times \log_{10} 50 = 2.303 \times (1 + \log 5) = 2.303 \times 1.699 \approx \mathbf{3.912}$