📌 What is a Logarithm?

A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must the base be raised to get a certain number?"

If \( b^y = x \), then \( \log_b x = y \)

Read as: "log base b of x equals y"

Conversion Rules:

📝 Examples - Exponential to Logarithmic:

Example 1: \( 2^3 = 8 \)

Base = 2, Exponent = 3, Result = 8
Logarithmic form: \( \log_2 8 = 3 \)

Example 2: \( 5^2 = 25 \)

Logarithmic form: \( \log_5 25 = 2 \)

Example 3: \( 10^{-2} = 0.01 \)

Logarithmic form: \( \log_{10} 0.01 = -2 \)

Example 4 (Rational Base): \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \)

Logarithmic form: \( \log_{1/2} \frac{1}{8} = 3 \)

📝 Examples - Logarithmic to Exponential:

Example 1: \( \log_3 9 = 2 \)

Exponential form: \( 3^2 = 9 \)

Example 2: \( \log_4 64 = 3 \)

Exponential form: \( 4^3 = 64 \)

Example 3: \( \log_{10} 1000 = 3 \)

Exponential form: \( 10^3 = 1000 \)

Definition:

A natural logarithm is a logarithm with base \( e \) (Euler's number ≈ 2.71828).

\( \ln x = \log_e x \)

📝 Examples - Natural Logarithms:

Example 1: \( e^3 = x \) → \( \ln x = 3 \)

Example 2: \( \ln 20 = y \) → \( e^y = 20 \)

Example 3: \( e^{-2} = 0.135 \) → \( \ln 0.135 = -2 \)

How to Evaluate:

To evaluate \( \log_b x = ? \), ask: "What power do I raise \( b \) to get \( x \)?"

📝 Examples - Evaluating:

Example 1: Evaluate \( \log_2 32 \)

Ask: "2 to what power = 32?"
\( 2^5 = 32 \)
Answer: \( \log_2 32 = 5 \)

Example 2: Evaluate \( \log_5 125 \)

\( 5^3 = 125 \)
Answer: \( \log_5 125 = 3 \)

Example 3: Evaluate \( \log_{10} 0.001 \)

\( 0.001 = \frac{1}{1000} = \frac{1}{10^3} = 10^{-3} \)
Answer: \( \log_{10} 0.001 = -3 \)

Example 4: Evaluate \( \log_3 \frac{1}{9} \)

\( \frac{1}{9} = \frac{1}{3^2} = 3^{-2} \)
Answer: \( \log_3 \frac{1}{9} = -2 \)

Example 5: Evaluate \( \ln e^4 \)

Using property \( \log_b b^n = n \)
Answer: \( \ln e^4 = 4 \)

Formula:

The change of base formula allows you to convert a logarithm from one base to another:

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

Where \( c \) is any positive base (commonly 10 or \( e \))

📝 Examples - Change of Base:

Example 1: Evaluate \( \log_2 10 \) using base 10

\( \log_2 10 = \frac{\log 10}{\log 2} = \frac{1}{0.301} \approx 3.322 \)

Example 2: Evaluate \( \log_5 20 \) using natural log

\( \log_5 20 = \frac{\ln 20}{\ln 5} = \frac{2.996}{1.609} \approx 1.861 \)

Example 3: Simplify \( \log_7 49 \)

Can solve directly: \( 7^2 = 49 \), so \( \log_7 49 = 2 \)
Or use change of base: \( \frac{\log 49}{\log 7} = \frac{1.690}{0.845} = 2 \)

Three Main Properties:

Additional Properties:

📝 Examples - Product Property:

Example 1: Expand \( \log_3 (5 \cdot 7) \)

\( \log_3 (5 \cdot 7) = \log_3 5 + \log_3 7 \)

Example 2: Condense \( \log_2 x + \log_2 y \)

\( \log_2 x + \log_2 y = \log_2 (xy) \)

Example 3: Simplify \( \log_5 10 + \log_5 25 \)

\( \log_5 10 + \log_5 25 = \log_5 (10 \cdot 25) = \log_5 250 \)

📝 Examples - Quotient Property:

Example 1: Expand \( \log_4 \frac{8}{3} \)

\( \log_4 \frac{8}{3} = \log_4 8 - \log_4 3 \)

Example 2: Condense \( \log_7 12 - \log_7 4 \)

\( \log_7 12 - \log_7 4 = \log_7 \frac{12}{4} = \log_7 3 \)

Example 3: Simplify \( \ln 100 - \ln 20 \)

\( \ln 100 - \ln 20 = \ln \frac{100}{20} = \ln 5 \)

📝 Examples - Power Property:

Example 1: Expand \( \log_2 x^5 \)

\( \log_2 x^5 = 5 \log_2 x \)

Example 2: Condense \( 3 \log_5 y \)

\( 3 \log_5 y = \log_5 y^3 \)

Example 3: Simplify \( \log_3 \sqrt{27} \)

\( \log_3 \sqrt{27} = \log_3 27^{1/2} = \frac{1}{2} \log_3 27 = \frac{1}{2} \cdot 3 = \frac{3}{2} \)

Example 4: Expand \( \ln x^{-2} \)

\( \ln x^{-2} = -2 \ln x \)

📝 Example 1 - Expand Completely:

Expand \( \log_2 \frac{x^3y}{z^2} \)

Step 1: Use quotient property

\( = \log_2 (x^3y) - \log_2 z^2 \)

Step 2: Use product property on first term

\( = \log_2 x^3 + \log_2 y - \log_2 z^2 \)

Step 3: Use power property

\( = 3\log_2 x + \log_2 y - 2\log_2 z \)

📝 Example 2 - Condense to Single Log:

Condense \( 2\log_5 x + 3\log_5 y - \log_5 z \)

Step 1: Use power property

\( = \log_5 x^2 + \log_5 y^3 - \log_5 z \)

Step 2: Use product property on first two terms

\( = \log_5 (x^2y^3) - \log_5 z \)

Step 3: Use quotient property

\( = \log_5 \frac{x^2y^3}{z} \)

📝 Example 3 - Evaluate Using Properties:

Evaluate \( \log_2 40 - \log_2 5 \)

Use quotient property:
\( = \log_2 \frac{40}{5} = \log_2 8 \)
Since \( 2^3 = 8 \):
\( = 3 \)

📝 Example 4 - Complex Evaluation:

If \( \log_3 2 = 0.631 \) and \( \log_3 5 = 1.465 \), find \( \log_3 10 \)

Note: \( 10 = 2 \times 5 \)
Use product property:
\( \log_3 10 = \log_3 (2 \times 5) = \log_3 2 + \log_3 5 \)
\( = 0.631 + 1.465 \)
\( = 2.096 \)

⚡ Quick Summary

• Logarithm is the inverse of exponentiation
• \( \log_b 1 = 0 \) and \( \log_b b = 1 \) always
• Use change of base for calculator evaluation
• Product → add logs; Quotient → subtract logs; Power → multiply coefficient
• To expand: break apart; To condense: combine into single log

📚 Common Logarithms Reference