Input Terms
Base must be positive and not equal to 1.
Max denom
| Sign | Coefficient k | Source base | Base value | Argument (symbolic) | Argument value (optional) | Remove |
|---|
Tip: Different source bases are allowed. The tool applies change-of-base to your chosen target base before condensing.
Example Data
Click “Load example” above to populate these rows.
| Sign | k | Source base | Base value | Argument (symbolic) | Argument value |
|---|---|---|---|---|---|
| + | 2 | 10 | 10 | x | 2 |
| + | 0.5 | e | — | y | 3 |
| - | 1 | 10 | 10 | z | 5 |
Condensed Result
Enter terms and click “Condense”.
Exponents display as fractions when enabled.
LaTeX (preview)
— generated after condensation —
Stepwise Derivation
Numeric Verification (optional)
Formula Used
Change of base: \(\log_b a = \dfrac{\ln a}{\ln b} = \dfrac{\log_B a}{\log_B b}\).
Power rule: \(k\,\log_b a = \log_b a^k\).
Product/Quotient: \(\log_b a + \log_b c = \log_b(ac)\), \(\log_b a - \log_b c = \log_b\!\left(\frac{a}{c}\right)\).
General condensation to base \(B\): \(\displaystyle\sum_i s_i k_i \log_{b_i} a_i = \log_{B}\!\Big(\prod_i a_i^{\,s_i k_i / \log_{B} b_i}\Big)\).
How to Use
- Choose the target base for the final single logarithm.
- Add each term with sign, coefficient \(k\), source base, and argument.
- Optionally supply numeric values of arguments for verification.
- Enable fractional exponents for tidy rational displays if desired.
- Click Condense, then copy the LaTeX if needed.
FAQs
Yes. The calculator converts each term to your chosen target base using the change-of-base formula before condensing.
Invalid. Logarithm base must be greater than 0 and not equal to 1. The tool flags improper inputs.
Yes. Symbolic text is supported for the final condensed form. Numeric checks require positive numeric values.
A negative sign multiplies the exponent. Negative exponents appear in the denominator of the condensed argument.
Enable fractional exponents to rationally approximate many decimal weights using continued fractions; the underlying calculations still use full precision.
A snapshot of the result, steps, LaTeX preview, and numeric verification in a print-ready layout suitable for sharing.