Calculation of Log Using Calculator

Logarithm Formula:

\[ \log_b(x) = \frac{\ln(x)}{\ln(b)} \]

Logarithm Result:

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1. What is Logarithm?

A logarithm is the inverse operation to exponentiation, meaning the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised to produce x.

2. How Does the Calculator Work?

The calculator uses the change of base formula:

\[ \log_b(x) = \frac{\ln(x)}{\ln(b)} \]

Where:

\( x \) — The positive real number whose logarithm is to be found
\( b \) — The positive real number base (must not be 1)
\( \ln \) — Natural logarithm (base e)

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

3. Importance of Logarithm Calculation

Details: Logarithms are fundamental in mathematics, science, and engineering. They are used in measuring sound (decibels), earthquake intensity (Richter scale), information theory, and solving exponential equations.

4. Using the Calculator

Tips: Enter positive values for both x and base (b). The base cannot be 1. For natural log (ln), set base to e (≈2.71828). For common log (base 10), leave base as 10.

5. Frequently Asked Questions (FAQ)

Q1: What is 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 happens if x is 0 or negative?
A: The logarithm is undefined for non-positive numbers in real numbers.

Q4: Why can't the base be 1?
A: The function would be constant (1^y always equals 1), making the inverse operation undefined.

Q5: How accurate is this calculation?
A: The calculation uses PHP's built-in log() function which provides high precision results.