Find The Log Calculator

Easily calculate the logarithm of a number to any base using our free Log Calculator.

x	logb(x)	ln(x)	log10(x)
1	0	0	0
2
e ≈ 2.718
10
Base (b)	1
100

Table of common logarithm values for the entered base.

What is a Log Calculator?

A Log Calculator is a tool used to find the logarithm of a given number with respect to a specified base. In mathematics, the logarithm (log) is the inverse operation to exponentiation, just as division is the inverse of multiplication. If you have b^y = x, then the logarithm of x to base b is y, written as log_b(x) = y.

For example, log₁₀(100) = 2 because 10² = 100. Our Log Calculator can handle any positive base (not equal to 1) and any positive number.

Who should use a Log Calculator?

• Students: Learning about logarithms in algebra, pre-calculus, and calculus often require using a Log Calculator for homework and understanding.
• Scientists and Engineers: Many scientific and engineering fields use logarithms to handle large ranges of numbers, such as in pH scales, decibel levels, and Richter magnitudes. A Log Calculator is essential here.
• Financial Analysts: Logarithms are used in finance for modeling growth rates and in certain financial formulas.
• Computer Scientists: Logarithms appear in the analysis of algorithms and data structures.

Common Misconceptions

One common misconception is that logarithms are only for very large or very small numbers. While they are useful for scaling such numbers, logarithms are fundamental mathematical functions used for numbers of all sizes greater than zero. Another is confusing natural logarithm (ln, base e) with common logarithm (log, base 10); our Log Calculator can handle both and other bases.

Log Calculator Formula and Mathematical Explanation

The fundamental relationship between exponentiation and logarithm is:

If b^y = x, then log_b(x) = y

Where:

• b is the base of the logarithm (must be b > 0 and b ≠ 1)
• x is the number whose logarithm is being taken (must be x > 0)
• y is the result (the exponent)

Most calculators, including the one in JavaScript, provide functions for the natural logarithm (ln, base e ≈ 2.71828) and sometimes the common logarithm (log, base 10). To calculate the logarithm to an arbitrary base 'b', we use the change of base formula:

log_b(x) = log_k(x) / log_k(b)

Where 'k' can be any base, typically 'e' (natural log) or 10. So, our Log Calculator uses:

log_b(x) = ln(x) / ln(b)

Variables Table

Variable	Meaning	Unit	Typical Range
b	Base	Dimensionless	b > 0, b ≠ 1
x	Number	Dimensionless	x > 0
y	Result (Logarithm)	Dimensionless	Any real number

Practical Examples (Real-World Use Cases)

Example 1: pH Scale

The pH of a solution is defined as the negative logarithm base 10 of the hydrogen ion concentration ([H⁺]). pH = -log₁₀([H⁺]). If a solution has a hydrogen ion concentration of 1 x 10^-4 moles per liter:

• Base (b) = 10
• Number (x) = 1 x 10^-4 = 0.0001
• Using the Log Calculator with base 10 and number 0.0001, we get log₁₀(0.0001) = -4.
• pH = -(-4) = 4. The solution is acidic.

Example 2: Decibel Scale

The decibel (dB) scale is used to measure sound intensity level and is logarithmic. The difference in decibels between two sound intensities I₁ and I₀ is L = 10 * log₁₀(I₁/I₀). If sound intensity increases 1000 times (I₁/I₀ = 1000):

• Base (b) = 10
• Number (x) = 1000
• Using the Log Calculator with base 10 and number 1000, log₁₀(1000) = 3.
• L = 10 * 3 = 30 dB increase.

You can use our Decibel calculator for more detailed calculations.

How to Use This Log Calculator

1. Enter the Base (b): Input the base of the logarithm you want to calculate in the "Base (b)" field. The base must be a positive number and not equal to 1. For natural logarithm (ln), you can enter 'e' or its approximate value 2.718281828459045.
2. Enter the Number (x): Input the number for which you want to find the logarithm in the "Number (x)" field. This number must be positive.
3. View Results: The Log Calculator will automatically display the result (log_b(x)), the formula used, and also show the natural log (ln(x)) and log base 10 (log₁₀(x)) of the number for reference. The chart and table will also update.
4. Reset: Click "Reset" to return the base to 10 and the number to 100.
5. Copy Results: Click "Copy Results" to copy the main result and input summary to your clipboard.

The results help you understand how many times the base needs to be multiplied by itself to get the number.

Key Factors That Affect Log Calculator Results

1. The Base (b): The value of the base significantly affects the result. A larger base means the logarithm grows more slowly. For a fixed number x > 1, log_b(x) decreases as b increases.
2. The Number (x): The value of the number directly influences the logarithm. For a fixed base b > 1, log_b(x) increases as x increases. If x is between 0 and 1, the logarithm is negative.
3. Base Being Close to 1: As the base gets very close to 1 (either from above or below), the absolute value of the logarithm becomes very large (approaching infinity), as the base raised to a very large positive or negative power is needed to get the number.
4. Number Being Close to 0: As the number approaches 0 (from the positive side), the logarithm (for base > 1) approaches negative infinity. Logarithms of zero or negative numbers are undefined in the real number system.
5. Base 'e' (Natural Logarithm): When the base is 'e' (approximately 2.71828), we are calculating the natural logarithm (ln). This is widely used in calculus and growth/decay models. Our math tools often involve natural logs.
6. Base 10 (Common Logarithm): When the base is 10, it's the common logarithm (log₁₀), often used in scales like pH and decibels due to our base-10 number system.

Frequently Asked Questions (FAQ)

What is a logarithm (log)?

The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. It's the inverse of exponentiation.

What is the natural logarithm (ln)?

The natural logarithm is the logarithm to the base 'e', where 'e' is Euler's number (approximately 2.71828). It's denoted as ln(x).

What is the common logarithm (log)?

The common logarithm is the logarithm to the base 10. It's often written as log(x) without explicitly stating the base 10, especially in scientific and engineering contexts.

Why can't the base of a logarithm be 1?

If the base were 1, 1 raised to any power is always 1. So, log₁(x) would only be defined if x=1 (and even then, it could be any value), and undefined for any other x. To have a unique and well-defined function, the base cannot be 1.

Why can't the base be negative or zero?

If the base were negative or zero, b^y would not behave consistently for non-integer values of y, leading to undefined or complex numbers for log_b(x). So, the base is restricted to positive numbers (not equal to 1).

Why must the number (x) be positive?

If the base 'b' is positive, b^y is always positive, regardless of the real number 'y'. Therefore, log_b(x) is only defined for positive 'x' in the real number system.

How do I calculate log base 2 using this Log Calculator?

Simply enter '2' in the "Base (b)" field and your desired number in the "Number (x)" field.

Is log_b(x) the same as ln(x)/ln(b)?

Yes, this is the change of base formula, and it's how our Log Calculator computes logarithms for any base using the natural logarithm function.

Related Tools and Internal Resources

• Exponent Calculator: Calculates the result of a number raised to a power, the inverse of the Log Calculator.
• Scientific Calculator: A comprehensive calculator that includes logarithmic and exponential functions.
• pH Calculator: Directly calculates pH from hydrogen ion concentration using log base 10.
• Decibel Calculator: Used for calculations involving the decibel scale, which is based on logarithms.
• Math Tools: Explore our collection of mathematical calculators.
• Science Calculators: Calculators for various scientific applications, some of which use logarithms.