1. What is a Logarithm?

A logarithm (base 10) answers the question: "To what power must 10 be raised to get this number?" For example, log₁₀(100) = 2 because 10² = 100. Logarithms transform multiplicative relationships into additive ones, making them useful for many scientific and engineering applications.

2. How to Calculate Logs

The base 10 logarithm is calculated using the formula:

Key Properties:

• \(\log_{10}(1) = 0\) (since 10⁰ = 1)
• \(\log_{10}(10) = 1\) (since 10¹ = 10)
• \(\log_{10}(a \times b) = \log_{10}(a) + \log_{10}(b)\)
• \(\log_{10}(a^n) = n \times \log_{10}(a)\)

3. Common Uses of Logarithms

Applications:

• Measuring earthquake intensity (Richter scale)
• Sound intensity (decibel scale)
• pH calculations in chemistry
• Data compression and signal processing
• Financial compound interest calculations

4. Calculator Instructions

How to Use:

1. Enter a positive number (x > 0) in the input field
2. Click "Calculate"
3. The result shows the power (y) to which 10 must be raised to get your number

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between log and ln?
A: log is base 10, while ln (natural log) is base e (≈2.718). Use ln for exponential growth/decay problems.

Q2: Can I calculate logs of negative numbers?
A: No, logarithms are only defined for positive real numbers in real number space.

Q3: How do I calculate logs with different bases?
A: Use the change of base formula: logₐ(b) = log₁₀(b)/log₁₀(a)

Q4: Why does log(0) give an error?
A: There's no finite power that 10 can be raised to get 0 (10^y approaches 0 as y approaches -∞).

Q5: How accurate is this calculator?
A: It uses PHP's built-in log10 function which provides about 14 decimal digits of precision.