1. What are Logarithmic Functions?

Logarithmic functions are the inverse of exponential functions. The base-10 logarithm (log₁₀) answers the question "10 raised to what power equals x?" while the natural logarithm (ln) with base e (≈2.718) answers "e raised to what power equals x?"

2. How Does the Calculator Work?

The calculator computes two logarithmic functions:

Where:

• \( x \) — Input value (must be positive)
• \( \log_{10}(x) \) — Base-10 logarithm (common log)
• \( \ln(x) \) — Natural logarithm (base e)

Explanation: The calculator uses PHP's built-in log10() and log() functions to compute these values with high precision.

3. Applications of Logarithms

Details: Logarithms are used in many scientific fields including:

• pH calculations in chemistry (logarithm of hydrogen ion concentration)
• Decibel scale in acoustics (logarithm of sound intensity)
• Richter scale for earthquake magnitude
• Compound interest calculations in finance
• Algorithm complexity in computer science

4. Using the Calculator

Tips: Enter any positive number to calculate its logarithms. The input must be greater than 0. Results are displayed with 4 decimal places.

5. Frequently Asked Questions (FAQ)

Q1: Why must the input be positive?
A: Logarithms are only defined for positive real numbers. The functions approach negative infinity as x approaches 0.

Q2: What's the difference between log and ln?
A: log (no base specified) typically means log₁₀ in many contexts, while ln always means natural log (base e). In programming, log() usually means natural log.

Q3: How do I calculate logarithms with other bases?
A: You can use the change of base formula: log_b(x) = log₁₀(x)/log₁₀(b) or ln(x)/ln(b).

Q4: What are the logarithms of special numbers?
A: log₁₀(1) = 0, ln(1) = 0, log₁₀(10) = 1, ln(e) = 1 where e ≈ 2.71828.

Q5: Can I calculate antilogarithms with this?
A: No, this calculator only computes logarithms. For antilogs, you would calculate 10^y or e^y.