Logarithm Calculator

Evaluate logarithms in any valid real base and connect the result to its inverse exponential relationship.

Calculation inputs

Precision & interpretation

Logarithms are generally irrational. Displayed decimals are rounded; the graph samples positive x values on a linear coordinate scale and marks the selected argument.

Calculation results

Logarithm value2log base 10 of 100

Natural log ln(x)4.605170186

Base10

CurveIncreasing

Step-by-step solution

Choose the formula
$logᵦ(x) = \frac{ln(x)}{ln(b)}$
Substitute your inputs
$log base 10 of 100 = \frac{ln(100)}{ln(10)}$
Calculate in order
$\frac{4.605170186}{2.302585093}$
Write the result
$y ≈ 2; 10^y = 100$

Logarithm curve

Sampled logarithm on positive x · linear axes. The marked point is your selected argument.

Floating-point evaluation; displayed decimals are rounded. No logarithm of a nonpositive real argument is returned.

Exact value & calculation details

Quantity	Value
Argument	100
Base	10
logᵦ(x)	2

Understand the calculation

Concept & meaning

A logarithm asks which exponent y makes bʸ equal x. In the real-number model, the argument must be positive and the base must be positive and different from 1.

Formula & special cases

Changing base reduces the calculation to ln(x)/ln(b). Bases larger than 1 produce increasing curves; bases between 0 and 1 produce decreasing curves.

Precision & limitations

Logarithms are generally irrational. Displayed decimals are rounded; the graph samples positive x values on a linear coordinate scale and marks the selected argument.

\log_{b} (x) = \frac{\ln (x)}{\ln (b)}

Worked example from your inputs

log base 10 of 100 = 2

Frequently asked questions

Can I use a base between 0 and 1?

Yes. The real logarithm is defined for 0 < b < 1 and x > 0, and the curve is decreasing.

Why is base 1 invalid?

Every real exponent of 1 is 1, so there is no one-to-one inverse logarithm.

Can the argument be zero or negative?

Not for a real logarithm. This calculator does not evaluate complex logarithm branches.

What is the difference between log and ln?

Common log uses base 10; ln uses base e. The base selector makes the convention explicit.

Show all questions (5 more)Show fewer questions

What does the Logarithm calculate?

Evaluate logarithms in any valid real base and connect the result to its inverse exponential relationship.

Which values should I enter?

Use the labeled fields above. x must be positive. Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

logᵦ(x) = ln(x) / ln(b); bʸ = x

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$OpenStax official logo$ OpenStax · Logarithmic propertiesLogarithm laws and change of baseRead the reference

References explain the mathematics; they do not imply institutional endorsement.