Logarithmic Functions and Change of Base

⚡ Key Takeaways

A logarithm answers the question "to what power must a base be raised to produce a given number?" — $\log_b(x) = y$ means $b^y = x$ . The change-of-base formula, $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$ , lets you rewrite any logarithm in terms of any other base, making all logarithms computable with a standard calculator. Logarithms appear everywhere exponential growth and decay occur: finance, acoustics (decibels), chemistry (pH), seismology (Richter scale), and information theory. The modern logarithm dates to John Napier (1614) and was refined by Henry Briggs into base-10 form.

💡 The Core Insight

A logarithm is an exponent in disguise. Every statement about logarithms can be translated back into a statement about exponents, and vice versa. The change-of-base formula exists because all exponential functions are related by a constant scaling of their exponents, so all logarithms are related by a constant scaling of their outputs.

DefinitionThe Logarithmic Function

For a fixed base $b$ with $b > 0$ and $b \neq 1$ , the logarithm base $b$ of a positive real number $x$ is the unique exponent $y$ satisfying:

1

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

Derivation hint: This is a definition, not a derivation — it simply names $\log_b$ as the inverse operation of raising $b$ to a power. Here:

$b$ is the base ( $b>0$ , $b\neq 1$ ),
$x$ is the argument ( $x > 0$ ),
$y = \log_b(x)$ is the logarithm (any real number).

The domain of $\log_b$ is $(0, \infty)$ and its range is all real numbers $(-\infty, \infty)$ .

DefinitionSpecial Bases and Notations

Three bases are used so often they have their own shorthand:

2

\quad \log_{10}(x) \equiv \log(x), \qquad \log_e(x) \equiv \ln(x), \qquad \log_2(x) \equiv \operatorname{lb}(x)

Derivation hint: These are naming conventions. "Common log" is base 10; "natural log" is base $e \approx 2.71828$ ; "binary log" is base 2, used in computer science.

TheoremFundamental Logarithm Laws

For $b>0$ , $b\neq 1$ , positive $M, N$ , and any real $p$ :

3

\quad \log_b(MN) = \log_b(M) + \log_b(N)

Derivation hint: Write $M=b^m$ , $N=b^n$ ; then $MN=b^{m+n}$ , so its log is $m+n$ .

4

\quad \log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

Derivation hint: Same substitution; $M/N = b^{m-n}$ .

5

\quad \log_b(M^p) = p\,\log_b(M)

Derivation hint: $M^p = (b^m)^p = b^{mp}$ , so its log is $mp = p\log_b(M)$ .

6

\quad \log_b(1) = 0, \qquad \log_b(b) = 1, \qquad b^{\log_b(x)} = x, \qquad \log_b(b^x) = x

Derivation hint: From the definition: $b^0=1$ , $b^1=b$ , and inverse functions cancel.

TheoremChange-of-Base Formula

For any valid bases $b$ and $c$ and any positive $x$ :

7

\quad \log_b(x) = \frac{\log_c(x)}{\log_c(b)}

Derivation hint: Take $\log_c$ of both sides of $b^{\log_b(x)} = x$ and apply law [5]. A useful special case: setting $x = c$ gives the reciprocal relation:

8

\quad \log_b(c) = \frac{1}{\log_c(b)}

Derivation hint: Put $x=c$ in [7]; the numerator $\log_c(c)=1$ .

ProofDeriving the Change-of-Base Formula

We prove equation [7].

Step 1. Let $y = \log_b(x)$ . By definition [1], this means

b^y = x.

Step 2. Apply the logarithm base $c$ to both sides. Because $\log_c$ is a function, equal inputs give equal outputs:

\log_c(b^y) = \log_c(x).

Step 3. Use the power law [5] on the left side, bringing the exponent $y$ out front:

y\,\log_c(b) = \log_c(x).

Step 4. Since $b \neq 1$ , we have $\log_c(b) \neq 0$ , so we may divide both sides by it:

y = \frac{\log_c(x)}{\log_c(b)}.

Step 5. Recall $y = \log_b(x)$ . Substituting back gives exactly equation [7]:

\log_b(x) = \frac{\log_c(x)}{\log_c(b)}. \qquad \blacksquare

Notice the proof uses only the definition and the power law — no special properties of any particular base are needed.

∎

mathematician · John Napier | 1550–1617

A Scottish landowner, theologian, and mathematician, John Napier spent roughly twenty years developing logarithms, publishing Mirifici Logarithmorum Canonis Descriptio in 1614. His goal was profoundly practical: to ease the brutal labor of multiplying long numbers in astronomy and navigation by converting multiplication into addition. Napier's original logarithms were not quite base $e$ or base 10 — they were defined through a clever kinematic analogy of two moving points — but they captured the essential additive magic. The astronomer Johannes Kepler used Napier's tables to speed his planetary calculations, famously saying logarithms had effectively doubled the life of the astronomer by halving their labor.

🧠 Intuition

Think of $\log_b(x)$ as asking: "How many times do I multiply $b$ by itself to reach $x$ ?" For base 10, $\log(1000)=3$ because $10\times10\times10 = 1000$ . For non-integer answers, the log smoothly interpolates: $\log(500) \approx 2.7$ , sitting between $\log(100)=2$ and $\log(1000)=3$ . The deepest intuition is that logarithms measure orders of magnitude, not raw size. Going from 10 to 100 and from 100 to 1000 each adds exactly 1 to the log, even though the second jump covers nine times more numerical distance. This is why logarithmic scales compress enormous ranges — a logarithmic axis treats "tenfold bigger" as a fixed, equal step. For the change-of-base formula, picture this: $\log_2(x)$ and $\log_{10}(x)$ both grow as $x$ grows, just at different rates. The formula says one is always a fixed multiple of the other — $\log_2(x) = \log_{10}(x) / \log_{10}(2) \approx 3.32 \cdot \log_{10}(x)$ . They are the same curve, vertically stretched.

Figure

Logarithmic curves for bases 2, e, and 10 — all passing through (1, 0)

All three curves cross the x-axis at $x=1$ (since $\log_b 1 = 0$ for every base) and pass through $(b, 1)$ . Smaller bases give steeper curves; they are constant multiples of one another.

ExampleComputing a Base-2 Logarithm with a Standard Calculator

Given: Find $\log_2(50)$ using a calculator that has only $\log$ (base 10) and $\ln$ buttons.

Step 1. Apply the change-of-base formula [7] with $b=2$ , $x=50$ , choosing base $c=10$ :

\log_2(50) = \frac{\log_{10}(50)}{\log_{10}(2)}.

Step 2. Evaluate each piece:

\log_{10}(50) \approx 1.69897, \qquad \log_{10}(2) \approx 0.30103.

Step 3. Divide:

\log_2(50) \approx \frac{1.69897}{0.30103} \approx 5.6439.

Step 4. Check with $\ln$ for confirmation: $\ln(50)/\ln(2) \approx 3.91202 / 0.69315 \approx 5.6439$ — identical, as the formula guarantees.

Sanity check: $2^5 = 32$ and $2^6 = 64$ , so $\log_2(50)$ must lie between 5 and 6. ✓

\boxed{\log_2(50) \approx 5.644}

ExampleSolving an Exponential Equation — Doubling Time

Given: A bacterial population doubles every cycle and follows $P(t) = 200 \cdot (1.15)^t$ , where $t$ is in hours. How long until the population reaches 1000?

Step 1. Set up the equation:

1000 = 200 \cdot (1.15)^t.

Step 2. Divide both sides by 200:

5 = (1.15)^t.

Step 3. Take the natural log of both sides and apply power law [5]:

\ln(5) = t\,\ln(1.15).

Step 4. Solve for $t$ (this is change-of-base in action, $t = \log_{1.15}(5)$ ):

t = \frac{\ln(5)}{\ln(1.15)} \approx \frac{1.60944}{0.13976} \approx 11.516 \text{ hours}.

Sanity check: $1.15^{11} \approx 4.65$ and $1.15^{12} \approx 5.35$ , so $t$ between 11 and 12 is correct. ✓

\boxed{t \approx 11.5 \text{ hours}}

⚠️ Common mistake

The single most common error is breaking up the logarithm of a sum or difference. There is NO law that simplifies $\log_b(M+N)$ . In particular:

\log_b(M+N) \neq \log_b(M) + \log_b(N).

The product law [3] applies to $\log_b(MN)$ , not $\log_b(M+N)$ . Likewise $\log_b(M-N)$ does not simplify. Other frequent traps:

$\dfrac{\log_b(M)}{\log_b(N)} \neq \log_b(M-N)$ — a ratio of logs is the change-of-base result, not a subtraction.
$(\log_b M)^2 \neq \log_b(M^2)$ — the first squares the output; the second uses power law [5] to give $2\log_b M$ .
The argument must be positive: $\log_b(0)$ and $\log_b(\text{negative})$ are undefined for real numbers.

Counterexample

Suppose a student claims $\log(M+N) = \log M + \log N$ . Test with $M=N=10$ :

Left side: $\log(10+10) = \log(20) \approx 1.301$ .
Right side: $\log(10) + \log(10) = 1 + 1 = 2$ .

Since $1.301 \neq 2$ , the claimed identity is false. (What is true is $\log(10 \cdot 10) = \log(100) = 2$ — the product law, not a sum law.)

connection

Logarithms are inseparable from their inverse, Exponential Functions: every property of one mirrors a property of the other, and graphing $y=\log_b x$ is just reflecting $y=b^x$ across the line $y=x$ . The natural logarithm $\ln$ is the gateway to Calculus of Exponentials and Logarithms, where $\frac{d}{dx}\ln x = \frac{1}{x}$ makes $\ln$ the most "natural" base. The product and quotient laws [3]–[4] are direct echoes of the Laws of Exponents, reflecting that logs convert multiplication to addition. Finally, the change-of-base formula connects to Inverse Functions, since it expresses how composing and inverting exponentials with different bases relate.

🔬 Where this appears

⏳ History

- ~1544 — Michael Stifel notes the correspondence between arithmetic and geometric progressions in Arithmetica Integra, foreshadowing logarithms.

- 1614 — John Napier publishes Mirifici Logarithmorum Canonis Descriptio, introducing logarithms to ease astronomical computation.

- 1617 — Henry Briggs visits Napier; together they conceive common (base-10) logarithms, far more convenient for decimal arithmetic.

- 1617–1624 — Briggs computes and publishes extensive base-10 log tables (Arithmetica Logarithmica), tabulating logs to 14 decimal places.

- 1620 — Edmund Gunter creates a logarithmic scale on a ruler; soon after, William Oughtred invents the slide rule, a logarithm-powered mechanical calculator used into the 1970s.

- 1647–1668 — Grégoire de Saint-Vincent and Nicholas Mercator link the area under the hyperbola

1/x

to logarithms, birthing the natural logarithm and the series for

\ln(1+x)

.

- 1683 — Jacob Bernoulli studies compound interest and encounters the constant

e

, later named for Euler.

- 1748 — Leonhard Euler, in Introductio in analysin infinitorum, systematizes logarithms as inverses of exponentials, fixes the notation

e

, and clarifies the role of the base.

open-problem

While real logarithms are fully understood, subtleties emerge at the boundaries:

Complex logarithms are multivalued. Extending $\log$ to complex numbers via $\log(re^{i\theta}) = \ln r + i(\theta + 2\pi k)$ yields infinitely many values, requiring a choice of "branch." This makes the simple law $\log(MN)=\log M + \log N$ fail on its principal branch for some complex inputs — a genuine pitfall studied in complex analysis.
Irrationality and transcendence. Determining whether a given logarithm is rational, irrational, or transcendental can be hard. The Gelfond–Schneider theorem (1934) settled some cases (e.g. $\log_2 3$ is irrational), but questions like whether $\log_2 3 + \log_3 2$ or certain combinations are transcendental connect to deep, partly open problems in transcendental number theory.
Computational precision. Although change-of-base makes any log computable, repeatedly dividing tiny numbers (e.g. $\log_b x$ for $b$ very close to 1) can amplify floating-point error — a practical limit in numerical computing.

Logarithmic Functions and Change of Base

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also