Logarithms Properties & Laws: Concepts

Definitions & Assumptions

Recall that if $a > 0$, $a \neq 1$, and $x > 0$, then: $$\log_a(x) = y \iff a^y = x$$

To ensure all proofs and identities are mathematically defined, we assume:

  • The base $a > 0$ and $a \neq 1$.
  • The numbers inside logarithms, $M$ and $N$, are positive real numbers ($M > 0$, $N > 0$).

Properties & Proofs

1. Product Rule

$\log_a(MN) = \log_a(M) + \log_a(N)$
Proof: Let $\log_a(MN) = x$, $\log_a(M) = y$, and $\log_a(N) = z$.
Converting to exponential form:
$a^x = MN$,  $a^y = M$,  and $a^z = N$.

Substitute $M$ and $N$ into the product:
$MN = a^y \cdot a^z = a^{y+z}$.
Since $a^x = MN$, we get $a^x = a^{y+z} \Rightarrow x = y + z$.

Replacing back the values:
$\log_a(MN) = \log_a(M) + \log_a(N)$.

2. Quotient Rule

$\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$
Proof: Let $\log_a(M) = y$ and $\log_a(N) = z$.
Converting to exponential form:
$a^y = M$  and $a^z = N$.

Divide the equations:
$\frac{M}{N} = \frac{a^y}{a^z} = a^{y-z}$.
Taking log on both sides:
$\log_a\left(\frac{M}{N}\right) = y - z = \log_a(M) - \log_a(N)$.

3. Power Rule

$\log_a(M^k) = k \log_a(M)$
Proof: Let $\log_a(M^k) = x$ and $\log_a(M) = y$.
$\Rightarrow a^x = M^k$  and $a^y = M$.

Substitute $M = a^y$ into $a^x = M^k$:
$a^x = (a^y)^k = a^{ky} \Rightarrow x = ky$.
Replacing values:
$\log_a(M^k) = k \log_a(M)$.

4. Change of Base Formula

$\log_a(M) = \frac{\log_b(M)}{\log_b(a)}$  (for $b > 0, b \neq 1$)
Proof: Let $\log_a(M) = x \Rightarrow a^x = M$.
Take log to base $b$ on both sides:
$\log_b(a^x) = \log_b(M) \Rightarrow x \log_b(a) = \log_b(M)$.
Solve for $x$:
$x = \frac{\log_b(M)}{\log_b(a)} \Rightarrow$ $\log_a(M) = \frac{\log_b(M)}{\log_b(a)}$.

5. Log of 1

$\log_a(1) = 0$
Proof: Let $\log_a(1) = x \Rightarrow a^x = 1$.
Since $a^0 = 1$ for any positive $a \neq 1$, we get $x = 0$.
Therefore, $\log_a(1) = 0$.

6. Log of a number to the same base

$\log_a(a) = 1$
Proof: Let $\log_a(a) = x \Rightarrow a^x = a$.
Since $a^1 = a$, we get $x = 1$.
Therefore, $\log_a(a) = 1$.

Summary of Logarithm Properties

Property Name Logarithmic Notation What it does
Product Rule $\log_a(MN) = \log_a(M) + \log_a(N)$ Turns multiplication into addition
Quotient Rule $\log_a(M/N) = \log_a(M) - \log_a(N)$ Turns division into subtraction
Power Rule $\log_a(M^k) = k \log_a(M)$ Exponents come down
Log of same base $\log_a(a) = 1$ Log of base itself is 1
Log of 1 $\log_a(1) = 0$ Logarithm of 1 to any base is 0
Base Changing $\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$ Changes base from $a$ to $b$
Important Warnings & Non-Identities

- $\log_a(m+n) \neq \log_a(m) + \log_a(n)$ (Logs do NOT distribute over addition!)
- $\log_a(m-n) \neq \log_a(m) - \log_a(n)$ (Logs do NOT distribute over subtraction!)
- If $x = y$, then $\log_a(x) = \log_a(y)$.

Ex 3
Example 3: Write as a single logarithm:

(a) $\log_7(3) + \log_7(5)$
(b) $\log_2(9) - \log_2(3)$
(c) $\log_4(3) + \log_4(6) - 3\log_4(2)$
(d) $\log_3(15) - \log_3(5)$

Show Solution
(a) $\log_7(3) + \log_7(5) = \log_7(3 \times 5) = \mathbf{\log_7(15)}$

(b) $\log_2(9) - \log_2(3) = \log_2\left(\frac{9}{3}\right) = \mathbf{\log_2(3)}$

(c) $\log_4(3) + \log_4(6) - \log_4(2^3) = \log_4(18) - \log_4(8) = \log_4\left(\frac{18}{8}\right) = \mathbf{\log_4\left(\frac{9}{4}\right)}$

(d) $\log_3(15) - \log_3(5) = \log_3\left(\frac{15}{5}\right) = \mathbf{\log_3(3)} = \mathbf{1}$
Ex 4
Example 4: Find the value of:

(a) $\log_7(343)$      (b) $\log_3(27\sqrt{3})$

Show Solution
(a) $\log_7(343) = \log_7(7^3) = 3\log_7(7) = 3(1) = \mathbf{3}$

(b) $27\sqrt{3} = 3^3 \cdot 3^{1/2} = 3^{3 + 1/2} = 3^{7/2}$.
$\Rightarrow \log_3(3^{7/2}) = \frac{7}{2}\log_3(3) = \mathbf{\frac{7}{2}}$
Ex 5
Example 5: Simplify: $\log_3(81) - \log_3(9)$.
Show Solution
Method 1: $\log_3(3^4) - \log_3(3^2) = 4 - 2 = \mathbf{2}$.
Method 2: $\log_3\left(\frac{81}{9}\right) = \log_3(9) = \log_3(3^2) = \mathbf{2}$.
Section 2.4

Common Logarithms (Base 10)

Logarithms with base 10 are called common logarithms. They are widely used in scientific scales, such as the Richter scale (earthquakes), pH scale (acidity), and decibel scale (sound loudness).

By convention, $\log_{10}(x)$ is written as simply $\log(x)$. When no base is written, base 10 is assumed.

Rules of Common Logarithms:

  • $\log(xy) = \log(x) + \log(y)$
  • $\log(x/y) = \log(x) - \log(y)$
  • $\log(x^n) = n\log(x)$
  • $\log(1) = 0$
  • $\log(10) = 1$
Ex 6
Example 6: Express as a single logarithm:

(a) $\log(2) + \log(7)$      (b) $\log(6) - \log(3)$

Show Solution
(a) $\log(2) + \log(7) = \log(2 \times 7) = \mathbf{\log(14)}$
(b) $\log(6) - \log(3) = \log\left(\frac{6}{3}\right) = \mathbf{\log(2)}$
Ex 7
Example 7: If $\log_3(7) = a$ and $\log_3(4) = b$, write the following in terms of $a$ and $b$:

(a) $\log_3\left(\frac{4}{7}\right)$      (b) $\log_3(28)$      (c) $\log_3\left(\frac{7}{3}\right)$

Show Solution
(a) $\log_3\left(\frac{4}{7}\right) = \log_3(4) - \log_3(7) = \mathbf{b - a}$

(b) $\log_3(28) = \log_3(4 \times 7) = \log_3(4) + \log_3(7) = \mathbf{b + a}$

(c) $\log_3\left(\frac{7}{3}\right) = \log_3(7) - \log_3(3) = a - 1 = \mathbf{a - 1}$
Ex 8
Example 8: Express as a single logarithm:

(a) $3\log(5) - 2$      (b) $1 + \log(2)$

Show Solution
(a) $3\log(5) - 2$:
Substitute $2 = 2\log(10) = \log(10^2) = \log(100)$.
$\Rightarrow \log(5^3) - \log(100) = \log(125) - \log(100) = \log\left(\frac{125}{100}\right) = \mathbf{\log\left(\frac{5}{4}\right)}$
*(Note: If the base is 2 instead of 10, i.e., $3\log_2(5) - 2 = \log_2(125) - \log_2(4) = \log_2(125/4)$). Let's follow the standard base-10 calculation.

(b) $1 + \log(2)$:
Substitute $1 = \log(10)$.
$\Rightarrow \log(10) + \log(2) = \log(10 \times 2) = \mathbf{\log(20)}$
Scaling Property & Natural Logarithms

- Common logarithms scale numbers that are very small or very large. For example, if quantities range from $0.0000000001$ to $10,000,000,000$, their common logarithms compress this huge range down to just $[-10, 10]$.
- Natural Logarithms use base $e$ ($e \approx 2.71828$, an irrational mathematical constant). They are denoted by $\ln(x)$, i.e., $\log_e(x) = \ln(x)$, and are fundamental in calculus.