1. What is a Logarithm?

A logarithm answers the question: "To what power must the base be raised to get this number?" The equation \(\log_b(x) = y\) means that \(b^y = x\). Logarithms are the inverse operations of exponentiation.

2. How to Solve Log Equations

To solve logarithmic equations without a calculator:

Steps:

1. Rewrite the logarithmic equation in exponential form
2. Express both sides with the same base when possible
3. Set exponents equal to each other and solve
4. For numerical solutions, use the change of base formula: \(\log_b(x) = \frac{\ln(x)}{\ln(b)}\)

3. Properties of Logarithms

Key Properties:

• Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
• Power Rule: \(\log_b(x^n) = n\log_b(x)\)
• Change of Base: \(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\) for any positive k ≠ 1

4. Common Log Bases

Important Bases:

• Base 10 (Common Log): \(\log_{10}(x)\) often written as \(\log(x)\)
• Base e (Natural Log): \(\log_e(x)\) written as \(\ln(x)\) where e ≈ 2.71828
• Base 2 (Binary Log): \(\log_2(x)\) common in computer science

5. Frequently Asked Questions (FAQ)

Q1: Can logarithms be negative?
A: The argument (x) must be positive, but the logarithm result (y) can be negative if 0 < x < 1.

Q2: What is log(0)?
A: Undefined. As x approaches 0 from the right, log(x) approaches -∞.

Q3: How to solve exponential equations using logs?
A: Take the log of both sides and use logarithm properties to solve for the variable.

Q4: What's the difference between log and ln?
A: log typically means log₁₀ (common log), while ln means logₑ (natural log with base e).

Q5: How to estimate logs without a calculator?
A: Use known log values and interpolation, or recognize powers of common bases.