Exponential, Logarithmic and Prime-Power Forms

Formula and method

Choose one of three modes. Factor a positive integer into prime powers, give a positive logarithm base and argument to find its exponent, or give a positive base and exponent to find its power. The natural base can be entered as e.

log_b(z)=y if and only if b^y=z, for b>0, b≠1 and z>0.

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• The three original modes and their controls remain. Logarithmic mode accepts base and argument; exponential mode accepts base and exponent. The captured offline inverse equations are retained, but hidden answer variables are not extra editable controls.
• For bounded exact prime factorization enter a positive integer at most 10000000000. One is the empty prime product. Factoring uses at most 50000 odd trial divisors and does not silently truncate large work.
• Real logarithm conversion requires a strictly positive base other than one and a strictly positive argument. Negative or zero bases are excluded because they do not have the promised equivalent real logarithm.
• This 64-digit implementation supports bases from 1e-6 through 1e6 at least 1e-12 away from one; exponent zero or magnitudes from 1e-6 through 10; and arguments from 1e-60 through 1e60, either exactly one or at least 1e-20 away from one. These are computational conditioning limits. Derived values outside them are withheld with guidance; near a computational boundary conservative rejection is possible.
• Decimal/e notation and bounded finite arithmetic must be exactly verifiable at every step. Repeating fractions, functions, general symbols and lossy expressions are rejected. Only the literal e on either base control is additionally recognized and stored as the natural constant rounded to 64 significant digits.
• Prime products are exact. Conversion notes show bases/arguments to 64 significant digits and exponents to 15 significant digits, with approximation signs; underlying decimal arithmetic uses 64 significant digits. Original input values display canonically without rounding a near-one base or argument to one. Complete conversion text is bounded to 2048 characters. Clearing required controls removes the result.
• With a known positive base other than one, argument one has exponent zero and exponent zero has argument one. Recovering an unknown base from exponent zero and argument one would be nonunique; the original interface does not expose that hidden inverse as a separate input mode.

Common questions

References

• Captured three-mode calculator and controls

Remove the exact captured Number-based recursive factorization and mantissa-formatting callback under its full helper hash and both original equation snapshots. The source-owned bounded integer factorization and local decimal conversion presenter replace its exports without executing captured JavaScript or valueMarkdown; all original algebraic branches are subsequently guarded by the separately pinned domain correction. Bind the complete three-mode exponential-form capture and complete compiler/served schemas to bounded local decimal logarithm/power and exact prime-factor models. Preserve all eight original variables, datasets, defaults and actual editable controls; retain six offline algebraic directions without exposing hidden l_exponent or e_number as external inputs. Repair zero log arguments/bases, unrelated normal_num group visibility, Number-based factorization/formatting and missing derived-domain explanations. Factor positive integers through 10000000000 with at most 50000 odd trial divisions and complete exact prime powers, representing one as the empty product. For real conversions use base 1e-6 through 1e6 at least 1e-12 from one, exponent zero or magnitude 1e-6 through 10, and positive argument 1e-60 through 1e60 either exactly one or at least 1e-20 from one; these are computational conditioning/work limits, not mathematical limits. Guard all six numerical directions before expensive power/exponential work and dynamically withhold unsupported derived answers through source-visible validity conditions. Accept only exact token e on e_base/l_base as the rounded 64-significant-digit natural-base constant through the existing source-bound exact-input hooks; all other symbols/functions remain disallowed and hidden/readonly API fields remain unwritable. Generate two declared bounded plain-text outputs and five small numeric summaries in the existing dispatcher, preserve full canonical input display and show conversion bases/arguments to 64 significant digits and exponents to 15 significant digits with approximation signs, and never execute captured JS, HTML or hotlinked engine code.

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