\log_b x=\frac{\log_k x}{\log_k b}Variables
- b: original base
- k: new base
- x: positive argument
How to use this formula
Rewrites a logarithm using another convenient base.
Important notes
- Bases must be positive and not equal to 1.
Quick example
log₂8=ln8/ln2=3.
Applicability, worked calculation, and verification
Domain and applicability
Apply Logarithm Change of Base only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- Bases must be positive and not equal to 1.
- For the Logarithm Change of Base, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.
- For the Logarithm Change of Base, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Logarithm Change of Base when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Logarithm Change of Base, check zero, negative, and extreme input values before relying on the result.
- When using Logarithm Change of Base, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \log_b x=\frac{\log_k x}{\log_k b} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Compute log₂8 as ln8/ln2. Since ln8=ln(2³)=3ln2, the quotient is 3.
- Choose k=e so the numerator is ln8 and the denominator is ln2.
- Use 8=2³ to rewrite ln8 as 3ln2.
- Cancel ln2 to obtain 3.
Independent verification
The defining relation 2³=8 confirms the logarithm.
Common mistakes
- When copying Logarithm Change of Base, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Logarithm Change of Base, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Logarithm Change of Base in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\log_b x=\frac{\log_k x}{\log_k b}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Logarithm Change of Base.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Logarithm Change of Base used for?
Rewrites a logarithm using another convenient base.
Can I copy this formula as LaTeX?
Yes. Copy \log_b x=\frac{\log_k x}{\log_k b} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.