Log Base 3 Of 9 Calculator With Solution

Logarithm Calculation Formula:

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

Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Logarithm?

A logarithm is the inverse operation to exponentiation, answering the question "to what power must the base be raised to produce a given number?" For example, log₃(9) = 2 because 3² = 9.

2. How to Calculate Log Base 3 of 9?

The calculator uses the logarithm change of base formula:

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

Example Calculation:

log₃(9) = ln(9)/ln(3)
ln(9) ≈ 2.1972
ln(3) ≈ 1.0986
2.1972 / 1.0986 ≈ 2

3. Importance of Logarithm Calculation

Applications: Logarithms are fundamental in mathematics, science, engineering, and computer science. They're used in measuring earthquake intensity (Richter scale), sound intensity (decibels), and in algorithms for efficient computation.

4. Using the Calculator

Instructions: Enter any positive number and base (≠1) to calculate the logarithm. The calculator will show the result and step-by-step solution.

5. Frequently Asked Questions (FAQ)

Q1: Why is ln used in the formula?
A: Natural logarithm (ln) can be used with any base via the change of base formula. Calculators typically have ln and log₁₀ functions built-in.

Q2: What if the base is 1?
A: Logarithm with base 1 is undefined because 1 raised to any power is always 1.

Q3: Can I calculate logarithms of negative numbers?
A: No, logarithms are only defined for positive real numbers in real number system.

Q4: What's the difference between ln and log?
A: ln is logarithm with base e (≈2.718), while log typically means log₁₀ unless specified otherwise.

Q5: How accurate is this calculation?
A: The calculation uses PHP's built-in log function which provides high precision (about 14 decimal digits).