Log Base Calculator
Compute logarithms for any positive base not equal to one. Enter the number and base to instantly see results and steps.
Enter Values
Equation Preview
Helping Notes
Use change of base to compute any base with natural or common logs.
If inputs are invalid (x ≤ 0, b ≤ 0, or b = 1), the result will show an error.
Results
Result Value
Change of Base (ln)
Change of Base (log10)
Exponent Form Check
What Is a Log Base Calculator?
A Log Base Calculator evaluates logarithms of positive numbers in arbitrary bases. Given a base and argument , it computes and explains each step using identities such as product, quotient, and power rules. You can switch freely among natural logarithms , common logs , and any custom base, then compare results side‑by‑side. This is useful in algebra, scientific notation, exponential growth/decay, information theory (bits), pH chemistry scales, sound intensity (decibels), and data transformations.
About the Log Base Calculator
The calculator relies on the change‑of‑base identity to evaluate logs in any base using a consistent backend. It enforces the domain conditions , , and , and it shows equivalences with exponential form for intuition. For exact powers it returns integers; for roots and rational powers it simplifies using the power rule; and for general inputs it provides precise decimals to your chosen precision. It can also express results in information units by choosing (bits), in scientific scales with , or in natural models with . Throughout, responsive formula rendering ensures legibility on phones and tablets.
Definition:
Change of base:
Product/Quotient:
Power rule:
Base conversions:
How to Use This Log Base Calculator
- Enter the argument and choose a base (e.g., ).
- Select output precision and whether to show steps using product/quotient/power rules or a direct change‑of‑base evaluation.
- Compute to see , the equivalent exponential form , and any simplifications.
- Optionally compare multiple bases at once and copy results to your notes or assignments.
Examples
- Exact power: since .
- Common log: .
- Root: because .
- Change of base: .
- Information measure: bits.
Formula Snippets Ready for Rendering
FAQs
What inputs are allowed?
The argument must be positive (). The base must be positive and not equal to one ().
When should I use base 10, base e, or base 2?
Use 10 for scientific notation, for calculus/growth models, and 2 for information measured in bits.
Why does the calculator use change of base?
It lets any base be computed from a consistent backend logarithm while preserving exact identities and numeric precision.
Can it simplify exact powers?
Yes—if , the result is the integer . Roots yield rational values via the power rule.
What happens if I enter zero or a negative argument?
Real logarithms are undefined there. The tool will warn and request a positive argument.
Does it support multiple bases simultaneously?
Yes—you can display together for quick comparisons.
How precise are the results?
Choose the number of decimal places; the calculator also retains exact symbolic forms when simplifications are possible.