Logarithm Calculator

Struggling with logarithmic calculations? Our Logarithm Calculator simplifies complex math! Whether you're working with natural logs (ln), base 10, or custom bases, get accurate results in seconds. Essential for algebra, chemistry, pH levels, and sound engineering. Just input your values and let our tool do the work.

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Logarithms: Calculation Formulas & Examples

1. Basic Logarithm Definition

Where:

• $b$ = Base (must be > 0 and ≠ 1)
• $a$ = Argument (must be > 0)
• $c$ = Exponent (the logarithm result)

Example:
log₂ (8) = 3     because 2³ = 8

2. Common Logarithm (Base 10)

Used in: pH calculations, decibel measurements
Example
log (100) = 2     because 10² = 100

3. Natural Logarithm (Base e)

e = 2.71828 (Euler's number)
Example
ln (20) = 3.0     because e^3.0 = 20.1

4. Change of Base Formula

Useful for: Calculating logarithms with uncommon bases
Example:

5. Logarithm Properties

Calculation Examples

Example 1: Solve for *x* in log₄ (x) = 3
4³ = x    So x = 64
Example 2: Simplify log₂ (8) + log ₂ (4)
log₂ (8 × 4) = log₂ (32) = 5
Example 3: Convert log₃ (7) to natural logarithms

100base and number 5 logarithms calculation example

Now, we will learn logarithms and logarithm calculation. For this, we take an example. Calculate the 100 base and number 5 logarithms calculation. Calculate the natural logarithm (loge) of the number 5:
log_e(5) = 1.609loge (5) = 1.609
Calculate the natural logarithm (loge) of the base 100:
log_e(100)≈4.605loge (100) = 4.605
Divide the result from step 1 by the result from step 2:

So, the correct value for log100 (5) is approximately 0.34950

Common logarithms (base 10). Below is a logarithm table for numbers 1 to 20.

The following rule can also be used to modify the logarithm's base.

Apply the following rule to swap the argument and base.

Other typical logarithms to be aware of are:

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