math calculator

Logarithm Calculator

Calculate logs for any custom base, common log (log10), and natural log (ln) with step-by-step rules.

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Inputs

Value (x)

Logarithm Base

Custom Base (b)

Results

log_base_10.00 (100)

2.00000000

Natural Log ln(x)4.60517019

Common Log log10(x)2.00000000

Binary Log log2(x)6.64385619

Step-by-Step Proof & Logic

1. Change of Base rule applied:

log_10.00(100) = ln(100) / ln(10.0000)

2. Evaluate natural logarithms:

ln(100) = 4.605170

ln(10.0000) = 2.302585

3. Perform division:

4.605170 / 2.302585 = 2.000000

Note: 100 is a perfect integer power of 10 (since 10^2 = 100).

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Formula

Logarithm Definition & Change of Base

log_b(x) = y  =>  b^y = x
log_b(x) = ln(x) / ln(b)

A logarithm is the power to which the base must be raised to yield x. Using natural logarithms, you can calculate the log for any custom base.

Step by step

How the calculation works

1Enter a positive value (x) and choose a base.
2Select Natural Log (e), Common Log (10), Binary Log (2), or enter a Custom positive base.
3The engine calculates natural logarithm ln(x) and natural logarithm of the base ln(b).
4Divide ln(x) by ln(b) to find the custom logarithm value.
5The proof explainer verifies if x is an exact integer power of the base.

Example

Custom Base Logarithm Example

To find log_4(16), we rewrite it as 4^y = 16. Since 4^2 = 16, the result is 2. The calculator also evaluates this as ln(16) / ln(4) = 2.772588 / 1.386294 = 2.0.

Related guides

Learn more about this calculation

Guide

Scientific Notation Guide: Powers of Ten Made Practical

Learn how to convert, compare, and calculate with powers of ten using worked examples and the live DTC scientific calculator.

Site Guide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

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Results

log_base_10.00 (100)

2.00000000

Natural Log ln(x)4.60517019

Common Log log10(x)2.00000000

Binary Log log2(x)6.64385619

Step-by-Step Proof & Logic

1. Change of Base rule applied:

log_10.00(100) = ln(100) / ln(10.0000)

2. Evaluate natural logarithms:

ln(100) = 4.605170

ln(10.0000) = 2.302585

3. Perform division:

4.605170 / 2.302585 = 2.000000

Note: 100 is a perfect integer power of 10 (since 10^2 = 100).

How to understand and use this calculator

The Definition and Mathematical Meaning of Logarithms

A logarithm is the mathematical inverse of exponentiation. It answers the question: "To what power must a base \(b\) be raised to yield a target number \(x\)?" If the relationship is written as \(b^y = x\), then the logarithm of \(x\) with base \(b\) is equal to \(y\).

Written algebraically:

\[\log_b(x) = y \iff b^y = x\]

Logarithms are used to solve equations where the unknown variable is located in the exponent. They are essential in algebra, chemistry (pH scales), geology (Richter scale), acoustics (decibels), computer science (algorithmic complexity), and biology.

The Change-of-Base Formula and Constants

Most numerical calculators only feature built-in functions for the natural logarithm (base \(e\)) and common logarithm (base 10). To compute a logarithm with a custom base \(b\), we apply the change-of-base formula:

\[\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}\]

This formula is derived by rewriting the exponential equation in logarithmic form and taking the log of both sides. It allows us to calculate any valid logarithm using natural logarithms.

Three logarithmic bases are particularly common:

Natural Logarithm (ln): Uses Euler's number \(e \approx 2.7182818\) as its base. It is the default log in calculus due to its derivative properties.

Common Logarithm (log): Uses base 10. It aligns with the decimal number system, making order-of-magnitude estimates straightforward.

Binary Logarithm (log2): Uses base 2. It is central to computer science and information theory, representing binary doubling steps.

Logarithmic Scales versus Linear Scales

A linear scale increases by adding a constant amount (e.g., 1, 2, 3, 4). A logarithmic scale increases by multiplying by a constant factor (e.g., 10, 100, 1000, 10000).

Logarithmic scales are used when dealing with data that spans several orders of magnitude. For example, the Richter scale measures earthquakes: a magnitude 6 earthquake is not 10% stronger than a magnitude 5; it is 10 times more powerful. Similarly, the decibel scale for sound and the pH scale for acidity are logarithmic, matching the way human senses perceive sound and chemical activity.

Related concepts

Change-of-Base Formula

The identity that converts any logarithm into a ratio of logs in another base.

Exponential Function

The inverse partner of the logarithm function.

Euler's Number e

A fundamental constant used in natural logs and continuous compounding equations.

Frequently Asked Questions

What is a logarithm?+

A logarithm is the power to which a base must be raised to yield a given number.

What is the natural logarithm?+

The natural logarithm, written ln(x), is the logarithm to the base e (Euler's constant, approximately 2.71828).

What is the common logarithm?+

The common logarithm, written log(x), is the logarithm to the base 10.

What is the change-of-base formula?+

It is log_b(a) = ln(a) / ln(b). It is used to calculate logarithms for custom bases.

Can you take the logarithm of a negative number?+

In real-number mathematics, the logarithm of a negative number or zero is undefined. This is because a positive base raised to any real power is always positive.

Why is log(1) always equal to 0?+

Because any base raised to the power of 0 equals 1: b^0 = 1.

Why is log_b(b) equal to 1?+

Because raising a base b to the power of 1 yields b: b^1 = b.

What is the inverse of a logarithm?+

Exponentiation is the inverse of logarithms. If y = log_b(x), the inverse is x = b^y.

How do you solve an equation with a variable in the exponent?+

Take the logarithm of both sides of the equation to bring the exponent down, then solve using standard algebra.

What is a logarithmic scale?+

A scale where each step is a multiple of the previous step (multiplicative growth) rather than a constant addition.

What is the derivative of ln(x)?+

In calculus, the derivative of the natural logarithm ln(x) with respect to x is 1/x.

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

Because 1 raised to any power is still 1. An equation like 1^y = 5 has no solution, and 1^y = 1 has infinite solutions, making base 1 mathematically useless for logarithms.

How the calculation works

1Enter a positive value (x) and choose a base.

2Select Natural Log (e), Common Log (10), Binary Log (2), or enter a Custom positive base.

3The engine calculates natural logarithm ln(x) and natural logarithm of the base ln(b).

4Divide ln(x) by ln(b) to find the custom logarithm value.

5The proof explainer verifies if x is an exact integer power of the base.

How to understand and use this calculator

The Definition and Mathematical Meaning of Logarithms

Written algebraically:

\[\log_b(x) = y \iff b^y = x\]

The Change-of-Base Formula and Constants

\[\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}\]

This formula is derived by rewriting the exponential equation in logarithmic form and taking the log of both sides. It allows us to calculate any valid logarithm using natural logarithms.

Three logarithmic bases are particularly common:

Natural Logarithm (ln): Uses Euler's number \(e \approx 2.7182818\) as its base. It is the default log in calculus due to its derivative properties.

Common Logarithm (log): Uses base 10. It aligns with the decimal number system, making order-of-magnitude estimates straightforward.

Binary Logarithm (log2): Uses base 2. It is central to computer science and information theory, representing binary doubling steps.

Logarithmic Scales versus Linear Scales

A linear scale increases by adding a constant amount (e.g., 1, 2, 3, 4). A logarithmic scale increases by multiplying by a constant factor (e.g., 10, 100, 1000, 10000).

Related concepts

Change-of-Base Formula

The identity that converts any logarithm into a ratio of logs in another base.

Exponential Function

The inverse partner of the logarithm function.

Euler's Number e

A fundamental constant used in natural logs and continuous compounding equations.

Frequently Asked Questions

What is a logarithm?+

A logarithm is the power to which a base must be raised to yield a given number.

What is the natural logarithm?+

The natural logarithm, written ln(x), is the logarithm to the base e (Euler's constant, approximately 2.71828).

What is the common logarithm?+

The common logarithm, written log(x), is the logarithm to the base 10.

What is the change-of-base formula?+

It is log_b(a) = ln(a) / ln(b). It is used to calculate logarithms for custom bases.

Can you take the logarithm of a negative number?+

In real-number mathematics, the logarithm of a negative number or zero is undefined. This is because a positive base raised to any real power is always positive.

Why is log(1) always equal to 0?+

Because any base raised to the power of 0 equals 1: b^0 = 1.

Why is log_b(b) equal to 1?+

Because raising a base b to the power of 1 yields b: b^1 = b.

What is the inverse of a logarithm?+

Exponentiation is the inverse of logarithms. If y = log_b(x), the inverse is x = b^y.

How do you solve an equation with a variable in the exponent?+

Take the logarithm of both sides of the equation to bring the exponent down, then solve using standard algebra.

What is a logarithmic scale?+

A scale where each step is a multiple of the previous step (multiplicative growth) rather than a constant addition.

What is the derivative of ln(x)?+

In calculus, the derivative of the natural logarithm ln(x) with respect to x is 1/x.

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

Because 1 raised to any power is still 1. An equation like 1^y = 5 has no solution, and 1^y = 1 has infinite solutions, making base 1 mathematically useless for logarithms.

Logarithm Calculator

Inputs

Results

Bookmark the Logarithm Calculator

Logarithm Definition & Change of Base

How the calculation works

Custom Base Logarithm Example

Learn more about this calculation

Scientific Notation Guide: Powers of Ten Made Practical

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Logarithm Calculator

Inputs

Results

Bookmark the Logarithm Calculator

Key output metrics

Logarithmic Curve for Base 10.00

How to understand and use this calculator

The Definition and Mathematical Meaning of Logarithms

The Change-of-Base Formula and Constants

Logarithmic Scales versus Linear Scales

Related concepts

Frequently Asked Questions

Logarithm Definition & Change of Base

How the calculation works

Custom Base Logarithm Example

Learn more about this calculation

Scientific Notation Guide: Powers of Ten Made Practical

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Key output metrics

Logarithmic Curve for Base 10.00

How to understand and use this calculator

The Definition and Mathematical Meaning of Logarithms

The Change-of-Base Formula and Constants

Logarithmic Scales versus Linear Scales

Related concepts

Frequently Asked Questions