The Instrument that
Conquered the Universe
The Slide Rule — Computing Before Computers
Before electronic calculators existed, scientists and engineers used a device called a slide rule — two rulers that could be slid against each other to multiply, divide, and compute powers. The slide rule was powered entirely by logarithms. It was used to design the Apollo spacecraft, calculate artillery trajectories in World War II, and build every skyscraper and bridge before 1970.
John Napier invented logarithms in 1614 specifically to reduce multiplication to addition — because $\log(ab) = \log a + \log b$. Astronomers at the time spent years doing multiplications by hand. Napier's logarithms reduced that work to weeks. The astronomer Laplace called logarithms "an admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer."
The Unfinished Business from §4.1
In §4.1, we computed the doubling time of an investment at 8% continuous compounding:
$$2P = Pe^{0.08T} \Rightarrow 2 = e^{0.08T} \Rightarrow T = \frac{\ln 2}{0.08} \approx 8.66 \text{ years}$$
We used $\ln$ without formally defining it. Now we fix that. Logarithms are the systematic answer to: "What power do I raise the base to, in order to get this number?"
Similarly: if PKR 50,000 grows to PKR 150,000 at 6% continuous compounding, how long does it take? The answer requires logarithms.
What Is a Logarithm?
The logarithm answers one specific question: what exponent gives me this number?
For $b > 0$, $b \neq 1$, and $x > 0$:
$$y = \log_b x \quad\Longleftrightarrow\quad b^y = x$$
Read: "y equals log base b of x" means "b raised to y equals x."
The logarithm $\log_b x$ is the exponent to which $b$ must be raised to get $x$. The domain is $(0,\infty)$ and the range is $(-\infty,\infty)$.
$\log_b x = y$ ⟺ $b^y = x$
These two equations say the exact same thing in two different languages. Every logarithm problem can be rewritten as an exponential and vice versa.
Evaluating Logarithms
Step by Step
Example 1 — Evaluate $\log_2 8$
Example 2 — Evaluate $\log_3 \frac{1}{27}$
Example 3 — Evaluate $\log_{25} 5$
Example 4 — Evaluate $\log_b 1$ for any valid base $b$
$\log_b 1 = 0$ (because $b^0 = 1$)
$\log_b b = 1$ (because $b^1 = b$)
The Laws of Logarithms
Exponential vs Logarithmic Rules — Side by Side
| Exponential Rule | Logarithmic Rule | |
|---|---|---|
| $b^x\cdot b^y = b^{x+y}$ | ↔ product becomes sum | $\log_b(MN)=\log_b M+\log_b N$ |
| $\dfrac{b^x}{b^y}=b^{x-y}$ | ↔ quotient becomes difference | $\log_b(M/N)=\log_b M-\log_b N$ |
| $(b^x)^y=b^{xy}$ | ↔ power becomes factor | $\log_b(M^p)=p\log_b M$ |
| $b^0=1$ | ↔ | $\log_b 1=0$ |
| $b^1=b$ | ↔ | $\log_b b=1$ |
Rewriting Log Expressions
Using the Rules
Example 5 — Expand $\log_3\!\left(\dfrac{x^2\sqrt{y}}{z^3}\right)$
Example 6 — Condense $3\ln x - \frac{1}{2}\ln y + 2\ln z$ into a single logarithm
Example 7 — Simplify $\log_4 2 + \log_4 8$
Example 8 — Expand $\log\sqrt[3]{\dfrac{a^4}{b^2c}}$
Example 9 — Solve $\log_2 x + \log_2(x-2) = 3$
Graphing Logarithms and
Their Relationship to Exponentials
Since $y=\log_b x$ is the inverse of $y=b^x$, their graphs are reflections of each other across the line $y=x$. Toggle the curves below to see this relationship clearly.
b > 1 — Increasing Log
As $x\to\infty$: $\log_b x\to\infty$ (slowly). As $x\to 0^+$: $\log_b x\to-\infty$. The y-axis is a vertical asymptote.
Examples: $\log_2 x$, $\log_{10} x$, $\ln x$
Key Point — All Share (1, 0)
Every logarithm $\log_b x$ passes through $(1,0)$ because $b^0=1$ for any base. This mirrors how every exponential passes through $(0,1)$.
The x-intercept of the log is at $x=1$, always.
Properties of
$f(x) = \log_b x$
| Property | Value / Description |
|---|---|
| Domain | $(0,\infty)$ — logarithm is only defined for positive inputs |
| Range | $(-\infty,\infty)$ — all real numbers |
| x-intercept | $(1,0)$ — because $\log_b 1=0$ for any $b$ |
| y-intercept | None — the y-axis is a vertical asymptote |
| Vertical asymptote | $x=0$ — the function approaches $-\infty$ as $x\to 0^+$ |
| Behaviour (b > 1) | Increasing — larger inputs give larger outputs |
| Behaviour (0 < b < 1) | Decreasing |
| One-to-one? | Yes — different inputs give different outputs |
| Inverse function | $f^{-1}(x)=b^x$ (the exponential function) |
| As $x\to+\infty$ | $\log_b x\to+\infty$ (slowly for $b>1$) |
| As $x\to 0^+$ | $\log_b x\to-\infty$ |
$\ln x$ — The Logarithm
Calculus Was Built For
The natural logarithm is the logarithm with base $e \approx 2.71828$:
$$\ln x = \log_e x \quad\Longleftrightarrow\quad e^y = x$$
$\ln x$ is the natural choice for calculus because $\frac{d}{dx}[\ln x] = \frac{1}{x}$ — the cleanest possible derivative. No other base gives such a clean formula.
Example 10 — Solve $e^{2x-1}=5$
Example 11 — Solve $3^x = 10$
Example 12 — Solve $2e^{3x}+1=9$
Converting Between
Different Bases
Most calculators only have $\log_{10}$ (log) and $\ln$ (natural log) buttons. The change of base formula lets you compute any logarithm using these.
$$\log_b x = \frac{\log_c x}{\log_c b} = \frac{\ln x}{\ln b}$$
where $c$ is any convenient new base (typically 10 or $e$). Top stays on top (the argument $x$), bottom stays on bottom (the original base $b$).
Example 13 — Compute $\log_5 200$ using a calculator
Example 14 — Compute $\log_3 50$ using $\log_{10}$
Example 15 — Solve $\log_6 x = 2.5$ for $x$
Using Logarithms
in Finance
Now that we have logarithms, we can solve compounding problems where the unknown is time — not the balance.
Finding the Time to Reach a Target
PKR 80,000 is invested at 7% compounded continuously. How long until it reaches PKR 200,000?
Finding the Required Interest Rate
You want PKR 500,000 to grow to PKR 1,000,000 in 10 years with continuous compounding. What rate is required?
Tripling Time
At 5% continuous compounding, how long does it take for an investment to triple?
Comparing Compounding Frequencies — Solving for Time
At 8% compounded monthly, how long for PKR 100,000 to become PKR 250,000?
How Fast Does
a Quantity Double?
For a quantity growing continuously as $Q(t) = Q_0 e^{kt}$ (with $k>0$):
$$T_{\text{double}} = \frac{\ln 2}{k}$$
Derivation: Set $Q(T)=2Q_0$:
$$2Q_0 = Q_0 e^{kT} \Rightarrow 2=e^{kT} \Rightarrow kT=\ln 2 \Rightarrow T=\frac{\ln 2}{k}$$
Similarly: tripling time $= \dfrac{\ln 3}{k}$, halving time (decay) $= \dfrac{\ln 2}{k}$ with $k<0$.
Example 20 — A bacterial population grows at rate $k=0.4$ per hour. Find the doubling time.
Example 21 — Pakistan's GDP grows at 4.5% per year continuously. When will it double?
Example 22 — A population grows from 5,000 to 8,000 in 6 years. Find $k$ and the doubling time.
Reading the Clock
Buried in Every Living Thing
The Discovery that Changed History
In 1949, American chemist Willard Libby developed radiocarbon dating — a technique that uses the known decay rate of $^{14}\text{C}$ (carbon-14) to determine the age of organic materials up to about 50,000 years old. Libby won the Nobel Prize in Chemistry in 1960.
Every living organism absorbs carbon from the atmosphere, including a small fixed ratio of radioactive $^{14}\text{C}$ alongside stable $^{12}\text{C}$. When the organism dies, it stops absorbing carbon. The $^{14}\text{C}$ already present begins decaying at a known, constant rate.
By measuring how much $^{14}\text{C}$ remains in a sample, we can compute how long ago the organism died. This has been used to date the Dead Sea Scrolls, Egyptian mummies, the Shroud of Turin, and the remains of ancient civilisations across the world.
The amount $Q(t)$ of a radioactive substance remaining at time $t$ satisfies:
$$Q(t) = Q_0 e^{-kt} \quad (k > 0)$$
where $Q_0$ is the initial amount and $k>0$ is the decay constant.
The half-life $T_{1/2}$ satisfies $Q(T_{1/2}) = Q_0/2$:
$$T_{1/2} = \frac{\ln 2}{k}$$
Carbon-14 has a half-life of approximately 5,730 years. So $k = \ln 2 / 5730 \approx 0.0001209$ per year.
Age of an Archaeological Sample
A piece of charcoal from an ancient fire site contains 72% of the $^{14}$C expected in living wood. How old is it? (Half-life of $^{14}$C $\approx 5{,}730$ years)
Remaining Activity
Polonium-210 has a half-life of 138 days. A sample initially has 50 mg. How much remains after 200 days?
Finding the Half-Life
A radioactive isotope decays from 80 g to 52 g in 40 years. Find the half-life.
Carbon Dating — The Dead Sea Scrolls
The Dead Sea Scrolls were tested and found to contain about 78% of the expected $^{14}$C. Estimate their age.