1. What is the Logarithmic Formula?

The logarithmic formula \(\log_b(a) = c\) represents the exponent \(c\) to which the base \(b\) must be raised to obtain the number \(a\). It's the inverse operation of exponentiation.

2. How Does the Calculator Work?

The calculator uses the logarithmic formula:

Where:

• \( a \) — The positive real number you want to find the logarithm of (a > 0)
• \( b \) — The positive real base of the logarithm (b > 0, b ≠ 1)
• \( c \) — The logarithm result (exponent)

Explanation: The logarithm answers the question "To what power must the base be raised to get this number?"

3. Importance of Logarithmic Calculation

Details: Logarithms are fundamental in mathematics and science, used in decibel scales, pH calculations, earthquake magnitude, and many algorithms in computer science.

4. Using the Calculator

Tips: Enter positive values for both a and b (b cannot be 1). The calculator will compute the exponent c that satisfies the equation b^c = a.

5. Frequently Asked Questions (FAQ)

Q1: What is the natural logarithm?
A: The natural logarithm (ln) is a logarithm with base e (Euler's number, approximately 2.71828).

Q2: What is the common logarithm?
A: The common logarithm is a logarithm with base 10, often written as log without a base specified.

Q3: Why can't the base be 1?
A: The function would be constant (1^c always equals 1), making the logarithm undefined for a ≠ 1.

Q4: What are logarithm properties?
A: Key properties include: log_b(xy) = log_b(x) + log_b(y), log_b(x/y) = log_b(x) - log_b(y), and log_b(x^y) = y·log_b(x).

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