🌲 Introduction to Logarithms

The Definition

$$\text{If } b^y = x, \text{ then } \log_b(x) = y$$

Translation:

• $\log_b(x)$ means "log base $b$ of $x$"
• It's asking: "$b$ to what power gives $x$?"

Examples

• $\log_2(8) = 3$ because $2^3 = 8$
• $\log_{10}(100) = 2$ because $10^2 = 100$
• $\log_5(25) = 2$ because $5^2 = 25$
• $\log_3(27) = 3$ because $3^3 = 27$

Key Insight: Logarithms are the INVERSE operation of exponentiation!

Just like subtraction undoes addition, logarithms undo exponentiation ✨

Common Logarithm (Base 10)

$\log(x)$ without a base written means $\log_{10}(x)$

• $\log(10) = 1$ because $10^1 = 10$
• $\log(100) = 2$ because $10^2 = 100$
• $\log(1000) = 3$ because $10^3 = 1000$
• $\log(1) = 0$ because $10^0 = 1$

Natural Logarithm (Base $e$)

$\ln(x)$ means $\log_e(x)$ where $e \approx 2.71828...$

The number $e$ is called Euler's number, fundamental in calculus!

• $\ln(e) = 1$ because $e^1 = e$
• $\ln(e^2) = 2$ because $e^2 = e^2$
• $\ln(1) = 0$ because $e^0 = 1$

💡 Calculator tip: Most calculators have "LOG" (base 10) and "LN" (base $e$) buttons!

The Three Main Properties

1. Product Property:

$$\log_b(xy) = \log_b(x) + \log_b(y)$$

Multiplication inside becomes addition outside!

Example: $\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$

2. Quotient Property:

$$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$$

Division inside becomes subtraction outside!

Example: $\log_3(27 \div 9) = \log_3(27) - \log_3(9) = 3 - 2 = 1$

3. Power Property:

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

An exponent inside can be pulled out front!

Example: $\log_2(8^3) = 3 \cdot \log_2(8) = 3 \cdot 3 = 9$

Strategy: Ask the Key Question

To find $\log_b(x)$, ask: "$b$ to what power equals $x$?"

Example 1: Find $\log_4(64)$

Ask: "$4$ to what power equals $64$?"

Try powers of 4: $4^1 = 4$, $4^2 = 16$, $4^3 = 64$ ✓

$$\log_4(64) = 3$$

Example 2: Find $\log_5(1)$

Ask: "$5$ to what power equals $1$?"

Any number to the zero power equals 1!

$$\log_5(1) = 0$$

Example 3: Find $\log_7(7)$

Ask: "$7$ to what power equals $7$?"

$$\log_7(7) = 1$$

Important Facts

• $\log_b(1) = 0$ for any base $b$ (because $b^0 = 1$)
• $\log_b(b) = 1$ for any base $b$ (because $b^1 = b$)
• $\log_b(b^x) = x$ (logarithm undoes exponentiation!)

The Formula

$$\log_b(x) = \frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}$$

This lets you find any logarithm with a calculator!

Example: Find $\log_2(10)$

Most calculators don't have base-2 logarithms, so use change of base:

$$\log_2(10) = \frac{\log(10)}{\log(2)} = \frac{1}{0.301} \approx 3.32$$

Example: Find $\log_5(50)$

$$\log_5(50) = \frac{\ln(50)}{\ln(5)} \approx \frac{3.912}{1.609} \approx 2.43$$

Expanding (Breaking Apart)

Example 1: Expand $\log_3(9x^2)$

$= \log_3(9) + \log_3(x^2)$ (product property)

$= 2 + 2\log_3(x)$ (power property) ✨

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

$= \log(x^3) - \log(y^2)$ (quotient property)

$= 3\log(x) - 2\log(y)$ (power property)

Condensing (Combining)

Example 1: Condense $2\log(x) + \log(y)$

$= \log(x^2) + \log(y)$ (power property)

$= \log(x^2y)$ (product property) ✨

Example 2: Condense $3\ln(x) - \ln(y)$

$= \ln(x^3) - \ln(y)$ (power property)

$= \ln\left(\frac{x^3}{y}\right)$ (quotient property)