PG2.3 The logarithm rules, proved

Simulate

Check the claim with numbers

Lesson 1.4.3 gave three log rules and checked them on a few numbers; here they meet 10,000 random cases. Each case picks a base $b$ , positive numbers $x$ and $y$ , and a power $k$ . It then measures how far each rule is from exact.

import math import random random.seed(3) worst = 0.0 for _ in range(10_000): b = random.choice([2, 10, math.e, 0.5]) x, y = random.uniform(0.01, 100), random.uniform(0.01, 100) k = random.uniform(-5, 5) gaps = [ math.log(x * y, b) - (math.log(x, b) + math.log(y, b)), math.log(x / y, b) - (math.log(x, b) - math.log(y, b)), math.log(x**k, b) - k * math.log(x, b), ] worst = max(worst, max(abs(g) for g in gaps)) print("every gap is below 1e-12:", worst < 1e-12)

⌘/Ctrl + Enter runs the cell

Every gap is below $10^{-12}$ , the size of rounding error in the last digits. The rules need positive inputs, though. The next cell tries $x = -3$ in the power rule, and a sum in place of a product.

import math print("ln 9 =", math.log(9), " 2 ln 3 =", 2 * math.log(3)) try: print(2 * math.log(-3)) except ValueError: print("ln(-3) does not exist") print("ln(2 + 3) =", math.log(2 + 3), " ln 2 + ln 3 =", math.log(2) + math.log(3))

⌘/Ctrl + Enter runs the cell

Since $(-3)^2 = 9$ , the left side $\ln\left((-3)^2\right)$ is $\ln 9 \approx 2.1972$ , but $2 \ln(-3)$ does not exist. And there is no rule for the log of a sum: 1.6094 against 1.7918.

Conjecture

b > 0

Ten thousand cases agree to within $10^{-12}$ . Rounding hides small failures, and no test can try every positive number, so the rules still need a proof.

Turn a log fact into an exponent fact

A logarithm is an exponent (lesson 1.4.3): $\log_b y$ is the power of $b$ that gives $y$ . So a fact about logs can be proved as a fact about powers, where the rules are known.

Rule: Reading off a log

\begin{gathered} b^w = y \ \Rightarrow \ \log_b y = w \\ (b > 0,\ b \ne 1,\ y > 0) \end{gathered}

For $b > 1$ the graph of $b^x$ always rises, and for $0 < b < 1$ it always falls (lesson 1.4.2). Either way it reaches each positive $y$ exactly once. So $\log_b y$ is one number, and showing $b^w = y$ proves that it is $w$ .

The proofs also use the exponent rules for real exponents. Lesson PG2.2 proved them for whole numbers. Lesson 1.4.2 extends powers of a positive base to every real exponent, with the same rules, and this stage takes that as known:

\begin{gathered} b^u \cdot b^v = b^{u+v} \qquad \frac{b^u}{b^v} = b^{u-v} \\ \left(b^u\right)^k = b^{uk} \end{gathered}

Format 2

Fill the gap

Here are the three proofs, each a line long once the logs are named. Fill each gap as you would a quick answer, such as u+v, then check.

u = \log_b x

How to type answers

Powers with ^, brackets round a power that is a sum: r^k, r^(k+1).
Fractions with / and brackets: (1-r)/(1+r). Multiply with * or side by side: 2k+1.
Press Enter in any box to check them all.

Format 3

Find the error

This proof claims that $\ln\left(x^2\right) = 2 \ln x$ for every $x \ne 0$ . The simulation found $\ln 9$ but no $2 \ln(-3)$ , so one line must be wrong. Select it, then say why.

Simulate

A claim of your own to prove

Lesson 1.4.3 also gave the change-of-base formula, $\log_b y = \frac{\ln y}{\ln b}$ . The cell computes $w = \frac{\ln y}{\ln b}$ for four pairs, then raises $b$ to the power $w$ . If $b^w = y$ , then $w$ is $\log_b y$ by its definition.

import math for b, y in [(2, 10), (10, 0.5), (0.5, 8), (7, 7)]: w = math.log(y) / math.log(b) print(f"b = {b}, y = {y}: w = {w:.6f}, b^w = {b**w:.6f}") try: math.log(5) / math.log(1) except ZeroDivisionError: print("b = 1: ln 1 = 0, so the formula divides by zero")

⌘/Ctrl + Enter runs the cell

In all four rows $b^w$ comes back to $y$ , for example $2^{3.321928} = 10$ . A base of 1 breaks the formula, because $\ln 1 = 0$ .

Conjecture

b > 0

Four pairs agree. Prove the formula for every base and every positive $y$ , using the power rule you have just proved.

Format 4

Write it

Prove the change-of-base formula: for $b > 0$ , $b \ne 1$ and $y > 0$ , show that $\log_b y = \frac{\ln y}{\ln b}$ . Say why the division is allowed.

Your proof

Saved in this browser as you type.

Write your own first: a proof you have tried to write teaches far more than one you only read.

u = \log_b x

The logarithm rules, proved

Check the claim with numbers

Turn a log fact into an exponent fact

Rule: Reading off a log

Fill the gap

Find the error

A claim of your own to prove

Write it

Summary card

Free further reading