Condense into a Single Logarithm Calculator

Simplify multiple logs into one exact expression. See rules, examples, graph output, and exportable results. Study smarter with neat answers for classwork and exams.

Enter matching logarithm terms, keep one base, and combine sums or differences into one clean logarithmic expression.

Calculator

Base Settings

Base must be positive and not equal to 1.

Graph Settings

Entry Rules

  • All terms must use the same base.
  • Use x for graphable variable expressions.
  • Use * for multiplication, like 3*x.
  • Arguments must stay positive on the chosen domain.
  • Coefficient multiplies the logarithm.
  • Inner power raises the argument inside the logarithm.

Term 1

Term 2

Term 3

Term 4

Reset

Formula Used

Product rule: logb(M) + logb(N) = logb(MN)
Quotient rule: logb(M) - logb(N) = logb(M/N)
Power rule: k·logb(M) = logb(Mk)
Inside power: logb(Mp) = p·logb(M)

This calculator combines all valid terms into one logarithm by multiplying the outside coefficient and the inside power for each term.

Positive combined exponents move to the numerator. Negative combined exponents move to the denominator. The final result stays under one logarithm with the same base.

How to Use This Calculator

  1. Choose one logarithm base for every term.
  2. Enter the operator, coefficient, argument, and inner power for each term.
  3. Use expressions like x + 3, 2*x, or (x+1)/2.
  4. Set a graph range if your arguments contain x.
  5. Press Condense Now to show the result above the form.
  6. Use the CSV button for tabular export or the PDF button after calculation.

Example Data Table

Original Expression Condensed Single Logarithm
log10(x) + log10(y) log10(xy)
log2(a) - log2(b) log2(a/b)
3·ln(m) + ln(n) ln(m3n)
2·log5(p) - 4·log5(q) log5(p2/q4)

FAQs

1. Can this combine logarithms with different bases?

No. Logarithms must share the same base before they can be condensed directly. Convert them first if the bases are different.

2. What happens to subtraction between logs?

Subtraction becomes division inside the single logarithm. Terms with negative combined exponents move to the denominator.

3. Why does the calculator ask for both coefficient and power?

A coefficient multiplies the whole logarithm. An inner power affects the argument inside the logarithm. Their product becomes the final exponent in the condensed form.

4. Can I use decimals or fractions?

Yes. Decimal coefficients, custom bases, and powers are supported. Keep arguments positive on the chosen graph interval.

5. Why is my graph missing some points?

A logarithm is only defined for positive arguments. The graph automatically skips x-values where any argument becomes zero, negative, or invalid.

6. Does the calculator simplify algebra inside each argument?

No. It condenses logarithmic structure, not full symbolic algebra. Expressions stay in the form you entered unless exponents move them between numerator and denominator.

7. Is natural logarithm supported?

Yes. Choose the natural base option to work with ln expressions. The same product, quotient, and power laws still apply.

8. What do the CSV and PDF downloads contain?

CSV exports the current calculation details in table form. PDF saves the result section shown above the form for quick sharing or printing.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.