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Log and Antilog Calculator

Enter a number and its base, and the smart log calculator will determine the log, natural log, and anti-log, with calculations shown.

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Logarithm Calculator

This smart logarithm calculator helps compute logarithms and inverse logarithms (antilogs) for any number and base.

What is a Logarithm?

In mathematics, a logarithm (log) is the inverse of exponentiation. The logarithm of a number x to a base b is the exponent y such that:

b^y = x

Or in logarithmic form:

y = log_b(x)

Logarithm Rules:

Product Rule: log_b(xy) = log_b(x) + log_b(y)
Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)
Power Rule: log_b(x^y) = y * log_b(x)
Change of Base Rule: log_c(x) = log_b(x) / log_b(c)

Common Types of Logarithms:

Common Logarithm: Base 10 (log₁₀), widely used in engineering and sciences.
Natural Logarithm: Base e (~2.71828), used frequently in calculus, physics, and math.
Binary Logarithm: Base 2 (log₂), common in computer science and data representation.

For the calculator, enter base 10 for common logs, 2 for binary logs, or leave the base blank for natural logs.

How to Use the Logarithm Calculator

To Calculate Logarithms:

Select "Log" from the drop-down menu.
Enter the number for which you want the logarithm.
Enter the base of the logarithm (or leave empty for natural log).
Click "Calculate" to get the result.

To Calculate Antilogarithms (Inverse Log):

Select "Antilog" from the drop-down menu.
Enter the exponent value.
Enter the base.
Click "Calculate" to get the antilog (inverse log) result.

Relationship Between Exponents and Logarithms

Exponents are the inverse of logarithms and vice versa. For example:

8² = 64 → log₈(64) = 2

Thus, antilog is essentially exponentiation.

FAQs

What is log on a calculator?

The "log" function computes logarithms. It helps determine the exponent needed for a base to reach a number.

How do you calculate logarithms?

Use log rules. Example: log(1000 * 100) = log(1000) + log(100) = 3 + 2 = 5 → antilog = 100,000.

How do you find log base 2 of a number?

Use the change of base formula: log₂(y) = log₁₀(y) / log₁₀(2)

What is log₁₀ equal to?

log₁₀(x) = log(x) = ln(x)/ln(10)

What is Antilog?

Antilog is the inverse function of a logarithm:

log_b(x) = y → antilog_b(y) = x

Example: log(39.2) = 1.5933 → antilog(1.5933) = 39.2

How to convert log to antilog?

Identify the base of the logarithm.
Raise the base to both sides: y = log_b(x) → bʸ = x
Solve for x.

Examples of Log → Antilog Conversion:

log₄(1/8) = -3 → 1/8 = 4⁻³
log₄(28) = 3 → 28 = 4³
log₃₈(7) = 1/2 → 7 = 38¹/²
log₆(2) = 1/3 → 2 = 6¹/³

Logarithm with Arbitrary Base

To calculate log with any base a:

logₐ(x) = ln(x) / ln(a) = log₁₀(x) / log₁₀(a)

You can also use base 10 or natural log calculators for this.

Final Words

This logarithm calculator is useful for K-12 education, algebra, calculus, probability, and other fields requiring exponent and log computations.

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