1. What is a Logarithm?

A logarithm is the inverse operation to exponentiation, answering the question: "To what power must the base be raised to produce a given number?" The logarithmic equation \( y = \log_b(x) \) means that \( b^y = x \).

2. How the Calculator Works

The calculator solves any variable in the logarithmic equation:

Where:

• \( y \) — The exponent (result)
• \( b \) — The base (must be positive and ≠ 1)
• \( x \) — The argument (must be positive)

Calculation Methods:

• For \( y \): \( y = \frac{\ln(x)}{\ln(b)} \)
• For \( b \): \( b = e^{\frac{\ln(x)}{y}} \)
• For \( x \): \( x = b^y \)

3. Logarithm Properties

Key Properties:

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

4. Using the Calculator

Instructions:

1. Select which variable you want to solve for (y, b, or x)
2. Enter the other two known values
3. Click "Calculate" to get the result

Note: All inputs must be positive numbers, and base cannot be 1.

5. Frequently Asked Questions (FAQ)

Q1: What are common logarithm bases?
A: Base 10 (common log), base e (natural log, ~2.718), and base 2 (binary log) are most common.

Q2: Why can't the base be 1?
A: \( 1^y \) always equals 1, so the function wouldn't be one-to-one or useful.

Q3: What's the difference between log and ln?
A: "log" typically means base 10, while "ln" means base e (natural logarithm).

Q4: Can logarithms be negative?
A: The result (y) can be negative, but the base (b) and argument (x) must be positive.

Q5: How are logarithms used in real life?
A: Used in decibel scales, pH calculations, earthquake magnitudes, and many scientific measurements.