1. What is Logarithm?

The logarithm (log) of a number is the exponent to which the base must be raised to produce that number. For example, log₂(8) = 3 because 2³ = 8.

2. How Does the Calculator Work?

The calculator uses the logarithm change of base formula:

Where:

• \( x \) — The positive real number you want to find the logarithm of
• \( b \) — The positive real base of the logarithm (cannot be 1)
• \( \ln \) — Natural logarithm (base e)

Explanation: The formula allows calculation of logarithms with any base using natural logarithms.

3. Importance of Logarithms

Details: Logarithms are fundamental in mathematics, science, and engineering. They help solve exponential equations, describe logarithmic scales (like pH and decibels), and appear in many natural phenomena.

4. Using the Calculator

Tips: Enter a positive number (x > 0) and a positive base (b > 0, b ≠ 1). The calculator will compute log_b(x) using the change of base formula.

5. Frequently Asked Questions (FAQ)

Q1: What is the natural logarithm?
A: The natural logarithm (ln) has base e (≈2.71828), a fundamental mathematical constant.

Q2: What is the common logarithm?
A: The common logarithm (log) typically refers to base 10, often used in scientific calculations.

Q3: Why can't the base be 1?
A: log₁(x) is undefined because 1 raised to any power is always 1, so no unique solution exists.

Q4: What about negative numbers?
A: Real-valued logarithms are only defined for positive real numbers. Complex logarithms can handle negatives.

Q5: How are logarithms used in real life?
A: They're used in measuring earthquake intensity (Richter scale), sound intensity (decibels), acidity (pH), and in algorithms for computer science.