Math
Condensing Logarithms Calculator
Type an expression such as 2 log x + log y - log z or (1/2) ln(x) - 3 ln(y), and this calculator combines it into one logarithm.
Coefficients become exponents, added logs multiply inside, subtracted logs divide, like terms are merged, and constants like the 1 in log_2(x) + 1 are absorbed as log_2(2).
Separate terms with + and -. Put coefficients in front (2 log x, (1/2) ln y). All logs must share one base.
Try an example
Result
Single logarithm
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- Original
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- Rules used
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- Numeric value
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Student quick launch
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Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
Condensing is expanding in reverse
The same three The exponent you would need to raise a base to in order to get a number. properties that expand a log can be read right to left to combine several logs into one. The order flips too: power rule first (The number multiplying a variable, like the 3 in 3x. become The small raised number saying how many times to multiply the base by itself.), then product and quotient rules (addition becomes multiplication, subtraction becomes division).
Step order the calculator follows
- Combine like terms: log x + log x becomes 2 log x.
- Rewrite any constant as a log of the The number being raised to a power, or the side a shape's height is measured from.: 1 = log_b(b), 2 = log(100) in base 10.
- Power rule: move every coefficient into the exponent of its argument.
- Product rule: multiply all the added arguments together.
- Quotient rule: divide by the product of all the subtracted arguments.
- If everything is numeric, simplify the inside and evaluate.
Fraction coefficients and roots
A coefficient of 1/2 becomes an exponent of 1/2, which is a square root: (1/2) ln x = ln(x^(1/2)) = ln(sqrt x). Negative coefficients send the argument to the The bottom number of a fraction; it says how many equal parts make one whole.: -3 ln y = -ln(y^3), so it divides.
Input syntax
| Write | Meaning |
|---|---|
| 2 log x, 2log(x), 2*log(x) | coefficient 2 |
| (1/2) ln y or 1/2 ln y | coefficient one half |
| n log_b z | symbolic coefficient n, base b |
| log(x^2) | an argument that already has a power |
| + 1 or - 3 | a constant to absorb into the log |
How to use it
- Type the sum or difference of logs. Coefficients go in front: 3 log x, (1/2) ln y, n log_b z.
- Use the same base for every term (all log, all ln, or all log_2).
- Read the single-log answer and open the work to see the power, product, and quotient rules applied in order.
How to read the answer
The answer is one logarithm whose argument is a product of the added pieces divided by the subtracted pieces, each raised to its former coefficient. If everything inside is numeric, the calculator also simplifies the inside and evaluates the log exactly when it can.
Common mistakes and edge cases
- Combining logs with different bases. Use the change of base formula first.
- Forgetting to move the coefficient into the exponent before multiplying: 2 log x + log y is log(x^2 y), not log(2xy).
- Putting subtracted terms in the The top number of a fraction; it counts how many parts you have.. Every minus becomes part of the denominator.
- Treating a constant as a separate term: 1 + log_2 x condenses to log_2(2x) because 1 = log_2 2.
Worked examples
Classic three-term
2 log x + log y - log z
Single logarithm
log(x^2 y / z)
Fraction and negative coefficients
(1/2) ln(x) - 3 ln(y) - ln(z)
Single logarithm
ln(x^(1/2) / (y^3 z))
Constant absorbed
log_2(x) + log_2(x) + 1
Single logarithm
log_2(x^2 2)
Fully numeric
log(4) + log(25) = log(100) = 2
Single logarithm
log(100)
Symbolic coefficient
n log_b(x) + log_b(y)
Single logarithm
log_b(x^n y)
Everything cancels
log x - log x = 0
Single logarithm
0
Mixed bases
log x + ln y cannot be combined
Single logarithm
Error
Frequently asked questions
What does it mean to condense a logarithm?+
To rewrite a sum or difference of logs as one single logarithm using the power, product, and quotient properties in reverse.
How do you write 2 log x + log y - log z as a single log?+
Move the 2 into an exponent: log(x^2). Add means multiply: log(x^2 y). Subtract means divide: log(x^2 y / z).
Can you condense logs with different bases?+
Not directly. Convert them to a common base first with the change of base formula, then condense.
What happens to a plain number like + 1 in the expression?+
It is rewritten as a log of the base: 1 = log_b(b). So log_2(x) + 1 = log_2(x) + log_2(2) = log_2(2x).
Why does a negative coefficient go to the denominator?+
Because -k log A = log(A^-k) = log(1 / A^k), and subtracting a log divides the argument.
Does condensing change the value?+
No. Both forms are equal for every allowed input; condensing just changes how the expression is written.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026