1. What is a Logarithm?

A logarithm is the inverse operation to exponentiation, meaning the logarithm of a number x with respect to base b is the exponent to which b must be raised to yield x. In other words, if \( y = \log_b(x) \), then \( b^y = x \).

2. How Does the Calculator Work?

The calculator uses the logarithm formula:

Where:

• \( x \) — The positive real number whose logarithm is being calculated
• \( b \) — The positive real base of the logarithm (must not be 1)
• \( \ln \) — The natural logarithm (logarithm with base e)

Explanation: The formula converts a logarithm of any base to a ratio of natural logarithms, which can be easily calculated.

3. Importance of Logarithms

Details: Logarithms are fundamental in mathematics, science, and engineering. They are used in:

• Measuring earthquake intensity (Richter scale)
• Measuring sound intensity (decibels)
• Computer science (algorithm complexity)
• Finance (compound interest calculations)
• Many areas of physics and chemistry

4. Using the Calculator

Tips:

• Enter positive values for both x and base
• The base cannot be 1
• For natural log (ln), use base ≈ 2.71828
• For common log (log10), use base 10

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between log and ln?
A: "log" typically refers to base 10 logarithm, while "ln" refers to natural logarithm (base e ≈ 2.71828).

Q2: Can the base be less than 0?
A: No, the base must be a positive real number not equal to 1.

Q3: What is log base 10 of 100?
A: 2, because 10^2 = 100.

Q4: What is the natural log of e?
A: 1, because e^1 = e.

Q5: Why can't the base be 1?
A: Because 1 raised to any power is always 1, so the function isn't well-defined.