Handle products, quotients, powers, coefficients, and custom bases for homework. See steps instantly with clarity. Export results, compare forms, and study logarithm rules confidently.
Use the standard logarithm identities when the base is valid, the base is positive, the base is not one, and every argument stays positive.
| Example | Mode | Input | Output |
|---|---|---|---|
| 1 | Expand | 2 × log((x²y³)/z) | 4 × log(x) + 6 × log(y) - 2 × log(z) |
| 2 | Condense | 2 × log(x) - 3 × log(y) + log(z) | log((x²z)/(y³)) |
| 3 | Expand | log((ab²)/(c³d)) | log(a) + 2 × log(b) - 3 × log(c) - log(d) |
| 4 | Condense | ln(m) + 4ln(n) - 2ln(p) | ln((mn⁴)/(p²)) |
Expanding logarithms helps when solving equations, differentiating, integrating, or comparing terms. Condensing logarithms helps when simplifying final answers and solving exponential models. This calculator focuses on symbolic structure, not numerical approximation.
Expand logarithms when a product, quotient, or exponent appears inside one logarithm. Expansion is useful for algebra steps, solving equations, and checking equivalent forms.
Condense logarithms when several terms share the same base and you want one compact expression. It is common in final answers and inverse-log equation work.
Yes. The product and quotient rules only combine logarithms with the same base. Mixing bases changes the meaning and should not be condensed directly.
Yes. A coefficient in front of a logarithm can move inside as an exponent. That is the power rule and it works both ways.
Because the quotient rule states that subtracting logarithms equals the logarithm of a quotient. Positive terms build the numerator, while subtraction terms build the denominator.
Yes. Every logarithm argument must be positive. The base must also be positive and cannot equal one. Those conditions must hold before and after rewriting.
No. Correct expansion and condensation preserve equivalence. They only rewrite the same logarithmic relationship in a different but mathematically equal form.
Yes. Enter base e to work with natural logarithms. The calculator will display them using ln notation in the generated expressions.
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