Math
Expanding Logarithms Calculator
Type a single logarithm such as log(x^2 y / z), ln(sqrt(x) / y^3), or log_2(8x), and this calculator rewrites it as a sum and difference of separate logs.
Each rule (product, quotient, power) is named on its own line, numeric parts like log_2(8) = 3 are evaluated exactly, and sums inside the log are flagged as unsplittable.
Use ^ for powers, / for division, sqrt( ) for square roots, and log_b for a custom base. Letters next to each other multiply.
Try an example
Result
Expanded form
—
- Original
- —
- Rules used
- —
- Numeric value
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
The three expansion rules
| Rule | Statement | Example |
|---|---|---|
| Product rule | log_b(A × B) = log_b A + log_b B | log(5x) = log 5 + log x |
| Quotient rule | log_b(A / B) = log_b A - log_b B | ln(x / 3) = ln x - ln 3 |
| Power rule | log_b(A^k) = k × log_b A | log(x^4) = 4 log x |
Order of operations when expanding
- Quotient rule first: split the The top number of a fraction; it counts how many parts you have. from the The bottom number of a fraction; it says how many equal parts make one whole..
- Product rule next: split each side into separate factors.
- Power rule last: bring each The small raised number saying how many times to multiply the base by itself. (including 1/2 for square roots) out front.
- Evaluate any purely numeric logs, such as log(1000) = 3 or ln(e) = 1.
What cannot be expanded
There is no rule for the The exponent you would need to raise a base to in order to get a number. of a sum or difference. log(x + 1) stays as log(x + 1), and log(x^2 - 9) can only be expanded after factoring the inside as (x - 3)(x + 3). The calculator marks such pieces so you do not invent a rule that does not exist.
Input syntax
| Write | Meaning |
|---|---|
| log(...) | base 10 |
| ln(...) | base e |
| log_2(...) or log2(...) | base 2 (any number or letter works) |
| x^3, x^(1/2), x^n | powers, fractional powers, symbolic powers |
| sqrt(x), cbrt(x) | square and cube roots |
| xy or x*y or x y | multiplication |
| x / (y z) | division; group the denominator with parentheses |
How to use it
- Type one logarithm with its argument in parentheses, for example log(x^3 / y).
- Use log for The number being raised to a power, or the side a shape's height is measured from. 10, ln for base e, and log_b or log2 for another base.
- Write powers with ^, roots with sqrt( ) or ^(1/2), and division with /.
- Read the expanded form and open the work to see which rule created each line.
How to read the answer
Each factor multiplied inside the log becomes its own log added together; each factor divided becomes a log subtracted; each exponent moves out front as a The number multiplying a variable, like the 3 in 3x.. Numeric pieces such as log(100) are replaced by their exact value when one exists.
Common mistakes and edge cases
- log(x + y) cannot be split into log x + log y. Only products, quotients, and powers expand.
- The exponent applies only to the factor it is attached to: log(x y^2) = log x + 2 log y, not 2 log x + 2 log y.
- A square root is a power of 1/2, so log sqrt(x) = (1/2) log x.
- Keep the whole denominator negative: log(x / (y z)) = log x - log y - log z.
Worked examples
Product, quotient, and power
log(x^2 y / z)
Expanded form
2 log(x) + log(y) - log(z)
Square root and grouped denominator
ln(sqrt(x) / (y z^3))
Expanded form
(1/2) ln(x) - ln(y) - 3 ln(z)
Numeric part evaluated
log_2(8x)
Expanded form
3 + log_2(x)
Sum inside cannot split
log((x+1)^2 / y)
Expanded form
2 log(x+1) - log(y)
Symbolic exponent
log_b(x^n y)
Expanded form
n log_b(x) + log_b(y)
Fully numeric
log(1000 / 10)
Expanded form
2
Negative exponent
ln(x^-2 y)
Expanded form
ln(y) - 2 ln(x)
Not a logarithm
x^2 has no log to expand
Expanded form
Error
Frequently asked questions
What does it mean to expand a logarithm?+
To rewrite one log of a product, quotient, or power as several simpler logs added or subtracted, with exponents moved out front as coefficients.
How do you expand log(x^2 y / z)?+
Quotient rule: log(x^2 y) - log z. Product rule: log x^2 + log y - log z. Power rule: 2 log x + log y - log z.
Can you expand log(x + y)?+
No. There is no property for the log of a sum. It stays log(x + y) unless the inside can be factored into a product.
How do you expand a log with a square root?+
Rewrite the root as a power of 1/2, then use the power rule: log sqrt(x) = log x^(1/2) = (1/2) log x.
Do the rules work for ln and log base 2?+
Yes. The product, quotient, and power rules hold for every base, as long as every log in the expression uses the same base.
Why does log_2(8x) become 3 + log_2(x)?+
Product rule gives log_2 8 + log_2 x, and log_2 8 is exactly 3 because 2^3 = 8.
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.
Related calculators
Condensing Logarithms Calculator
Condense a sum or difference of logarithms into a single logarithm, applying the power, product, and quotient rules in order, combining like terms, and evaluating exact numeric results.
Logarithm Calculator
Calculate logarithms in any valid base with common log, natural log, and change-of-base steps.
Change of Base Formula Calculator
Evaluate log_b(x) in any base with the change of base formula, showing both the ln and log₁₀ routes, exact answers when x is a power of b, and every step.
Natural Log Calculator
Calculate the natural logarithm ln(x) or its inverse e^x, with exact answers for powers of e, the matching log₁₀ and log₂ values, series terms, and step-by-step work.
Log Base 2 Calculator
Calculate log base 2 of any positive number with an exact answer for powers of 2, the decimal value otherwise, and the bits and doublings interpretation.
Adding Fractions Calculator
Add up to five fractions or mixed numbers. Finds the least common denominator with prime factors, rewrites each fraction, adds, simplifies, and shows the mixed-number and decimal forms with a fraction-strip picture.
Last updated: September 4, 2026