Precalculus condensing logarithms worksheet

The same three rules read backwards. A coefficient goes up as a power first, then sums become products and differences become quotients.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Condensing Logarithms Date Period Condense each expression to a single logarithm. $9 \log_{5} (u) - 9 \log_{5} (n)$ $9 \log_{5} (m) - 5 \log_{5} (b)$ $6 \log_{2} (a) - 5 \log_{2} (v)$ $6 \log_{4} (x) - \log_{4} (b)$ $6 \log_{5} (x) + 6 \log_{5} (b)$ $7 \log_{7} (u) + 7 \log_{7} (n)$ $7 \log_{3} (x) - 4 \log_{3} (v)$ $6 \log_{2} (m) - 3 \log_{2} (n)$ $9 \log_{5} (x) - 7 \log_{5} (v)$ $9 \log_{3} (m) + 7 \log_{3} (y)$ $6 \ln (x) - 8 \ln (v)$ $8 \ln (m) + 2 \ln (y)$ $6 \ln (u) - 4 \ln (y)$ $8 \log_{4} (a) - 5 \log_{4} (n)$ $6 \log_{7} (x) + 3 \log_{7} (b)$ $8 \log_{3} (u) - 3 \log_{3} (n)$ $8 \log_{2} (u) - 6 \log_{2} (b)$ $9 \log_{5} (m) + 3 \log_{5} (v)$ $6 \ln (x) + \ln (v)$ $6 \log_{3} (m) + 3 \log_{3} (y)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Condensing Logarithms

Date Period

Condense each expression to a single logarithm.

$9 \log_{5} (u) - 9 \log_{5} (n)$
$9 \log_{5} (m) - 5 \log_{5} (b)$
$6 \log_{2} (a) - 5 \log_{2} (v)$
$6 \log_{4} (x) - \log_{4} (b)$
$6 \log_{5} (x) + 6 \log_{5} (b)$
$7 \log_{7} (u) + 7 \log_{7} (n)$
$7 \log_{3} (x) - 4 \log_{3} (v)$
$6 \log_{2} (m) - 3 \log_{2} (n)$
$9 \log_{5} (x) - 7 \log_{5} (v)$
$9 \log_{3} (m) + 7 \log_{3} (y)$
$6 \ln (x) - 8 \ln (v)$
$8 \ln (m) + 2 \ln (y)$
$6 \ln (u) - 4 \ln (y)$
$8 \log_{4} (a) - 5 \log_{4} (n)$
$6 \log_{7} (x) + 3 \log_{7} (b)$
$8 \log_{3} (u) - 3 \log_{3} (n)$
$8 \log_{2} (u) - 6 \log_{2} (b)$
$9 \log_{5} (m) + 3 \log_{5} (v)$
$6 \ln (x) + \ln (v)$
$6 \log_{3} (m) + 3 \log_{3} (y)$



Sigma Prep · sigmaprep.io/worksheets Name Condensing LogarithmsAnswer keyVersion 1 Date Period Condense each expression to a single logarithm. $9 \log_{5} (u) - 9 \log_{5} (n)$ $\log_{5} (\frac{u^{9}}{n^{9}})$ $9 \log_{5} (m) - 5 \log_{5} (b)$ $\log_{5} (\frac{m^{9}}{b^{5}})$ $6 \log_{2} (a) - 5 \log_{2} (v)$ $\log_{2} (\frac{a^{6}}{v^{5}})$ $6 \log_{4} (x) - \log_{4} (b)$ $\log_{4} (\frac{x^{6}}{b})$ $6 \log_{5} (x) + 6 \log_{5} (b)$ $\log_{5} (x^{6} b^{6})$ $7 \log_{7} (u) + 7 \log_{7} (n)$ $\log_{7} (u^{7} n^{7})$ $7 \log_{3} (x) - 4 \log_{3} (v)$ $\log_{3} (\frac{x^{7}}{v^{4}})$ $6 \log_{2} (m) - 3 \log_{2} (n)$ $\log_{2} (\frac{m^{6}}{n^{3}})$ $9 \log_{5} (x) - 7 \log_{5} (v)$ $\log_{5} (\frac{x^{9}}{v^{7}})$ $9 \log_{3} (m) + 7 \log_{3} (y)$ $\log_{3} (m^{9} y^{7})$ $6 \ln (x) - 8 \ln (v)$ $\ln (\frac{x^{6}}{v^{8}})$ $8 \ln (m) + 2 \ln (y)$ $\ln (m^{8} y^{2})$ $6 \ln (u) - 4 \ln (y)$ $\ln (\frac{u^{6}}{y^{4}})$ $8 \log_{4} (a) - 5 \log_{4} (n)$ $\log_{4} (\frac{a^{8}}{n^{5}})$ $6 \log_{7} (x) + 3 \log_{7} (b)$ $\log_{7} (x^{6} b^{3})$ $8 \log_{3} (u) - 3 \log_{3} (n)$ $\log_{3} (\frac{u^{8}}{n^{3}})$ $8 \log_{2} (u) - 6 \log_{2} (b)$ $\log_{2} (\frac{u^{8}}{b^{6}})$ $9 \log_{5} (m) + 3 \log_{5} (v)$ $\log_{5} (m^{9} v^{3})$ $6 \ln (x) + \ln (v)$ $\ln (x^{6} v)$ $6 \log_{3} (m) + 3 \log_{3} (y)$ $\log_{3} (m^{6} y^{3})$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Condensing LogarithmsAnswer keyVersion 1

Date Period

Condense each expression to a single logarithm.

$9 \log_{5} (u) - 9 \log_{5} (n)$ $\log_{5} (\frac{u^{9}}{n^{9}})$
$9 \log_{5} (m) - 5 \log_{5} (b)$ $\log_{5} (\frac{m^{9}}{b^{5}})$
$6 \log_{2} (a) - 5 \log_{2} (v)$ $\log_{2} (\frac{a^{6}}{v^{5}})$
$6 \log_{4} (x) - \log_{4} (b)$ $\log_{4} (\frac{x^{6}}{b})$
$6 \log_{5} (x) + 6 \log_{5} (b)$ $\log_{5} (x^{6} b^{6})$
$7 \log_{7} (u) + 7 \log_{7} (n)$ $\log_{7} (u^{7} n^{7})$
$7 \log_{3} (x) - 4 \log_{3} (v)$ $\log_{3} (\frac{x^{7}}{v^{4}})$
$6 \log_{2} (m) - 3 \log_{2} (n)$ $\log_{2} (\frac{m^{6}}{n^{3}})$
$9 \log_{5} (x) - 7 \log_{5} (v)$ $\log_{5} (\frac{x^{9}}{v^{7}})$
$9 \log_{3} (m) + 7 \log_{3} (y)$ $\log_{3} (m^{9} y^{7})$
$6 \ln (x) - 8 \ln (v)$ $\ln (\frac{x^{6}}{v^{8}})$
$8 \ln (m) + 2 \ln (y)$ $\ln (m^{8} y^{2})$
$6 \ln (u) - 4 \ln (y)$ $\ln (\frac{u^{6}}{y^{4}})$
$8 \log_{4} (a) - 5 \log_{4} (n)$ $\log_{4} (\frac{a^{8}}{n^{5}})$
$6 \log_{7} (x) + 3 \log_{7} (b)$ $\log_{7} (x^{6} b^{3})$
$8 \log_{3} (u) - 3 \log_{3} (n)$ $\log_{3} (\frac{u^{8}}{n^{3}})$
$8 \log_{2} (u) - 6 \log_{2} (b)$ $\log_{2} (\frac{u^{8}}{b^{6}})$
$9 \log_{5} (m) + 3 \log_{5} (v)$ $\log_{5} (m^{9} v^{3})$
$6 \ln (x) + \ln (v)$ $\ln (x^{6} v)$
$6 \log_{3} (m) + 3 \log_{3} (y)$ $\log_{3} (m^{6} y^{3})$

How to do these

$4 \log u - 2 \log v$

Coefficients go up as powers first.
That is log u to the fourth minus log v squared.
A difference becomes a quotient.

The answer is $\log (\frac{u^{4}}{v^{2}})$ .

Where students go wrong

Combining before moving the coefficients up. 4 log u is log of u to the fourth, not 4 times whatever is inside, and condensing in the wrong order gives an answer that is off by a power.

Also called combining logarithms or writing as a single logarithm.

Questions about these worksheets

Yes. Every condensing logarithms sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the condensing logarithms questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Combining before moving the coefficients up. 4 log u is log of u to the fourth, not 4 times whatever is inside, and condensing in the wrong order gives an answer that is off by a power.

Precalculus condensing logarithms worksheet

Select your difficulty

Condensing Logarithms

Condensing LogarithmsAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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