How To Find Logarithm On Calculator

Your expert tool for understanding and calculating logarithms.

Common Logarithm Examples

This table shows results for common logarithmic calculations.

Expression	Base (b)	Number (x)	Result (y)
log₂(8)	2	8	3
log₁₀(100)	10	100	2
logₑ(e²)	e ≈ 2.718	e² ≈ 7.389	2
log₅(1)	5	1	0

What is a Logarithm?

A logarithm is essentially the inverse operation of exponentiation. For instance, if we ask “what power do we need to raise 2 to get 8?”, the answer is 3 (since 2³ = 8). The logarithm expresses this relationship: log₂(8) = 3. In general, the equation logₛ(x) = y is the same as asking “to what power ‘y’ must the base ‘b’ be raised to get the number ‘x’?”, or bʸ = x. Logarithms are incredibly useful in science, engineering, and finance for simplifying calculations with large numbers and analyzing exponential growth.

The Logarithm Formula and Explanation

The core formula for a logarithm is:

logₛ(x) = y ⇔ bʸ = x

Most calculators, including the one on this page, don’t have a button for every possible base. They typically have a ‘log’ button (base 10) and an ‘ln’ button (base ‘e’ ≈ 2.718). To find the logarithm for any other base, we use the Change of Base Formula. This powerful rule lets us convert a logarithm from one base to another, most commonly base ‘e’ (the natural log).

logₛ(x) = ln(x) / ln(b)

Variables Table

Variable	Meaning	Unit	Typical Range
x	Argument or Number	Unitless	Any positive real number (x > 0)
b	Base	Unitless	Any positive real number not equal to 1 (b > 0 and b ≠ 1)
y	Logarithm	Unitless	Any real number

Practical Examples

Example 1: Common Logarithm

Let’s find the value of log₁₀(1000).

• Inputs: Number (x) = 1000, Base (b) = 10
• Question: 10 to what power equals 1000?
• Result: 3. Because 10³ = 1000.

Example 2: Using the Change of Base Formula

Suppose you want to calculate log₄(64) and your calculator only has ‘ln’. Check out our natural log vs common log guide for more details.

• Inputs: Number (x) = 64, Base (b) = 4
• Formula: log₄(64) = ln(64) / ln(4)
• Calculation: ln(64) ≈ 4.15888, and ln(4) ≈ 1.38629
• Result: 4.15888 / 1.38629 ≈ 3. Because 4³ = 64.

How to Use This Logarithm Calculator

1. Enter the Number (x): In the first field, type the number you wish to find the logarithm of. This value must be positive.
2. Enter the Base (b): In the second field, enter the base. This must also be a positive number and cannot be 1.
3. View the Result: The calculator automatically computes the answer and displays it in the result area. You don’t even need to click a button! It shows the result, an explanation in exponential form, and the intermediate calculation using natural logs.
4. Analyze the Graph: The chart below the calculator visualizes the logarithmic function for the base you selected, helping you understand its behavior.

Key Factors That Affect a Logarithm

• The Base (b): The base determines the rate of growth of the logarithmic curve. A base larger than 1 results in an increasing function, while a base between 0 and 1 results in a decreasing function.
• The Number (x): The value of the logarithm is directly dependent on this number. As ‘x’ increases (for b > 1), its logarithm also increases.
• The Domain: Logarithms are only defined for positive numbers (x > 0). You cannot take the log of zero or a negative number in the real number system.
• Base Value of 1: A base of 1 is not allowed because any power of 1 is still 1, making it impossible to get any other number. Learning the logarithm formula is key.
• Relationship between Base and Number: When the number (x) equals the base (b), the logarithm is always 1 (e.g., log₅(5) = 1).
• Number Value of 1: The logarithm of 1 is always 0, regardless of the base (e.g., log₅(1) = 0), because any base raised to the power of 0 is 1.

Frequently Asked Questions (FAQ)

1. How do you find the logarithm on a standard calculator?

Most scientific calculators have a “LOG” button for base 10 and an “LN” button for base e (natural log). To find log base ‘b’ of ‘x’, you use the change of base formula: `log(x) / log(b)` or `ln(x) / ln(b)`.

2. What is the difference between log and ln?

“Log” usually implies the common logarithm (base 10), while “ln” stands for the natural logarithm (base e ≈ 2.718). Both are fundamental in mathematics, with ‘ln’ being prevalent in calculus and ‘log’ in fields like chemistry (pH scale) and acoustics (decibels). For more info, see this exponent calculator.

3. Why can’t you take the log of a negative number?

In the real number system, it’s impossible. A positive base raised to any real power can never result in a negative number. For example, 2⁻² = 1/4, not -4. Therefore, the domain of a standard logarithm function is restricted to positive numbers.

4. What is the log of 1?

The logarithm of 1 to any valid base is always 0. This is because any positive number (not equal to 1) raised to the power of 0 equals 1 (e.g., b⁰ = 1).

5. What is an antilog?

An antilog is the inverse of a logarithm. If logₛ(x) = y, then the antilog of y (base b) is x. It’s the same as exponentiation: bʸ = x. An antilog calculator essentially finds the result of raising the base to the power of the logarithm.

6. What is the change of base rule?

The change of base rule allows you to rewrite a logarithm in terms of logs with a different base. The formula is logₛ(x) = log꜀(x) / log꜀(b). This is extremely useful for calculators that only compute logs in base 10 or e. The rule is explained well by the e-value calculator.

7. Are the input values unitless?

Yes. For a standard mathematical logarithm, both the base and the number (argument) are considered pure, unitless numbers.

8. How does this calculator handle edge cases?

This tool validates your input in real-time. If you enter a non-positive number or a base that is non-positive or equal to 1, it will display an error message and will not perform the calculation, as these are mathematically undefined.