Free Condense Logarithms Calculator

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Condensing logarithms means taking a sum or difference of several separate logarithm terms and collapsing them into a single logarithm of one combined expression — essentially running the standard logarithm rules in reverse from how they're usually first introduced. The product rule (log a + log b = log(ab)) turns addition of logs into multiplication inside a single log, and the quotient rule (log a − log b = log(a/b)) turns subtraction into division inside a single log. Condensing is a routine step whenever an equation with multiple log terms needs to be solved — you generally can't isolate a variable while it's trapped inside several separate logarithms, but once condensed into one logarithm, the equation can be solved directly by exponentiating both sides. This shows up throughout algebra and precalculus courses, and the same product/quotient rules underlie practical formulas in acoustics (combining decibel levels), chemistry (working with pH and other log-scale measurements), and information theory.

How it works

Enter a shared base and up to three logarithm terms connected by plus or minus signs. The calculator applies the product rule to every term joined by a plus sign — multiplying their arguments together inside a single logarithm — and the quotient rule to every term joined by a minus sign, dividing that term's argument out of the running product instead. The result is one single logarithm, of the same base, whose argument is built up from the appropriate combination of multiplication and division of the original arguments.

Enter base.
Enter first term (a) in log(a).
Enter operation before second term.
Enter second term (b), optional.
Enter operation before third term.
Enter third term (c), optional.
Click Calculate to see your results.

Examples

log(2) + log(3)

Using the product rule: log(2) + log(3) = log(2×3) = log(6).

log(20) − log(4)

Using the quotient rule: log(20) − log(4) = log(20/4) = log(5).

log(2) + log(3) − log(4)

Combining both rules in sequence: log(2)+log(3) becomes log(6) via the product rule, and then log(6) − log(4) becomes log(6/4) = log(3/2) via the quotient rule.

Common mistakes to avoid

Adding the arguments directly (log a + log b ≠ log(a+b)) instead of correctly multiplying them together inside the log.
Mixing terms from different logarithm bases as if they could be condensed together directly, without first converting to a common base.
Applying the quotient rule backward — subtracting the arguments (log a − log b ≠ log(a) − log(b) treated as log(a−b)) instead of dividing them.
Forgetting to apply the power rule first to clear a coefficient (like the 3 in 3·log(x)) before attempting to condense multiple log terms together.

Frequently asked questions

Yes — every term being combined must share exactly the same base. The product and quotient rules only hold within a single, consistent base; you can't directly condense log₂(a) + log₃(b) into one logarithm without first converting one of them to match the other's base.

Because that's not one of the actual logarithm rules — it's one of the most common algebra mistakes. The genuine product rule combines the arguments by multiplication, log(ab), not addition; log(a+b) has no simple relationship to log a and log b individually.

Condensing becomes essential once you need to solve an equation where the variable is trapped inside multiple separate log terms — collapsing them into one logarithm lets you exponentiate both sides of the equation to isolate the variable, which isn't possible while several logs remain unmerged.

Yes — the power rule states that a coefficient in front of a log can be moved inside as an exponent on the argument: 3·log(x) = log(x³). This tool focuses on combining separately added or subtracted log terms, but the power rule is often applied first to clear any coefficients before condensing.

Yes — the product and quotient rules apply identically to natural logarithms, since ln is simply a logarithm with base e; ln(a) + ln(b) = ln(ab) follows exactly the same pattern as any other base.

The underlying rules extend to any number of terms — you can chain the product and quotient rules across as many log terms as needed, applying them one pair at a time until everything collapses into a single combined logarithm; this tool specifically supports up to three terms at once for simplicity.