Condensing logarithms worksheet for Precalculus

Precalculus condensing mixes common logs, natural logs and other bases, with fraction coefficients that turn into roots. Coefficients become powers first, then plus terms multiply and minus terms divide, leaving a single logarithm, the form you need before solving a log equation.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Condensing Logarithms Date Period Condense each expression to a single logarithm. Condense into a single logarithm $8 \log_{7} (a) + 8 \log_{7} (b)$ $\frac{1}{2} \log (x) + 2 \log (y) - \frac{1}{3} \log (z)$ $8 \ln (m) + 4 \ln (v)$ $\frac{2}{3} \log (x) - 4 \log (y) - \frac{1}{3} \log (z)$ $9 \log_{5} (a) - 5 \log_{5} (y)$ $\log (x) - \frac{1}{2} \log (y) + \frac{1}{3} \log (z)$ $6 \ln (a) - 8 \ln (n)$ $\frac{1}{2} \log (x) + \frac{3}{2} \log (y) - 4 \log (z)$ $6 \log_{5} (m) + \log_{5} (v)$ $3 \log (x) - \frac{1}{3} \log (y) + \frac{3}{2} \log (z)$ $7 \ln (x) - \ln (y)$ $\frac{2}{3} \log (x) - \frac{1}{3} \log (y) - 2 \log (z)$ $6 \log_{2} (x) - 5 \log_{2} (n)$ $\frac{2}{3} \log (x) - \frac{1}{2} \log (y) + 2 \log (z)$ $7 \ln (u) + 3 \ln (b)$ $\frac{1}{3} \log (x) + \log (y) - \frac{1}{2} \log (z)$ $8 \log_{7} (u) - \log_{7} (y)$ $\log (x) - \frac{1}{2} \log (y) - \frac{3}{2} \log (z)$ $8 \ln (a) + \ln (v)$ $4 \log (x) - \frac{2}{3} \log (y) - \frac{1}{2} \log (z)$ 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.

Condense into a single logarithm
$8 \log_{7} (a) + 8 \log_{7} (b)$
$\frac{1}{2} \log (x) + 2 \log (y) - \frac{1}{3} \log (z)$
$8 \ln (m) + 4 \ln (v)$
$\frac{2}{3} \log (x) - 4 \log (y) - \frac{1}{3} \log (z)$
$9 \log_{5} (a) - 5 \log_{5} (y)$
$\log (x) - \frac{1}{2} \log (y) + \frac{1}{3} \log (z)$
$6 \ln (a) - 8 \ln (n)$
$\frac{1}{2} \log (x) + \frac{3}{2} \log (y) - 4 \log (z)$
$6 \log_{5} (m) + \log_{5} (v)$
$3 \log (x) - \frac{1}{3} \log (y) + \frac{3}{2} \log (z)$
$7 \ln (x) - \ln (y)$
$\frac{2}{3} \log (x) - \frac{1}{3} \log (y) - 2 \log (z)$
$6 \log_{2} (x) - 5 \log_{2} (n)$
$\frac{2}{3} \log (x) - \frac{1}{2} \log (y) + 2 \log (z)$
$7 \ln (u) + 3 \ln (b)$
$\frac{1}{3} \log (x) + \log (y) - \frac{1}{2} \log (z)$
$8 \log_{7} (u) - \log_{7} (y)$
$\log (x) - \frac{1}{2} \log (y) - \frac{3}{2} \log (z)$
$8 \ln (a) + \ln (v)$
$4 \log (x) - \frac{2}{3} \log (y) - \frac{1}{2} \log (z)$



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

Sigma Prep · sigmaprep.io/worksheets

Name

Condensing LogarithmsAnswer keyVersion 2

Date Period

Condense each expression to a single logarithm.

Condense into a single logarithm
$8 \log_{7} (a) + 8 \log_{7} (b)$ $\log_{7} ⁣ (a^{8} b^{8})$
$\frac{1}{2} \log (x) + 2 \log (y) - \frac{1}{3} \log (z)$ $\log ⁣ (\frac{\sqrt{x} y^{2}}{\sqrt[3]{z}})$
$8 \ln (m) + 4 \ln (v)$ $\ln ⁣ (m^{8} v^{4})$
$\frac{2}{3} \log (x) - 4 \log (y) - \frac{1}{3} \log (z)$ $\log ⁣ (\frac{\sqrt[3]{x^{2}}}{y^{4} \sqrt[3]{z}})$
$9 \log_{5} (a) - 5 \log_{5} (y)$ $\log_{5} ⁣ (\frac{a^{9}}{y^{5}})$
$\log (x) - \frac{1}{2} \log (y) + \frac{1}{3} \log (z)$ $\log ⁣ (\frac{x \sqrt[3]{z}}{\sqrt{y}})$
$6 \ln (a) - 8 \ln (n)$ $\ln ⁣ (\frac{a^{6}}{n^{8}})$
$\frac{1}{2} \log (x) + \frac{3}{2} \log (y) - 4 \log (z)$ $\log ⁣ (\frac{\sqrt{x} \sqrt{y^{3}}}{z^{4}})$
$6 \log_{5} (m) + \log_{5} (v)$ $\log_{5} ⁣ (m^{6} v)$
$3 \log (x) - \frac{1}{3} \log (y) + \frac{3}{2} \log (z)$ $\log ⁣ (\frac{x^{3} \sqrt{z^{3}}}{\sqrt[3]{y}})$
$7 \ln (x) - \ln (y)$ $\ln ⁣ (\frac{x^{7}}{y})$
$\frac{2}{3} \log (x) - \frac{1}{3} \log (y) - 2 \log (z)$ $\log ⁣ (\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{y} z^{2}})$
$6 \log_{2} (x) - 5 \log_{2} (n)$ $\log_{2} ⁣ (\frac{x^{6}}{n^{5}})$
$\frac{2}{3} \log (x) - \frac{1}{2} \log (y) + 2 \log (z)$ $\log ⁣ (\frac{\sqrt[3]{x^{2}} z^{2}}{\sqrt{y}})$
$7 \ln (u) + 3 \ln (b)$ $\ln ⁣ (u^{7} b^{3})$
$\frac{1}{3} \log (x) + \log (y) - \frac{1}{2} \log (z)$ $\log ⁣ (\frac{\sqrt[3]{x} y}{\sqrt{z}})$
$8 \log_{7} (u) - \log_{7} (y)$ $\log_{7} ⁣ (\frac{u^{8}}{y})$
$\log (x) - \frac{1}{2} \log (y) - \frac{3}{2} \log (z)$ $\log ⁣ (\frac{x}{\sqrt{y} \sqrt{z^{3}}})$
$8 \ln (a) + \ln (v)$ $\ln ⁣ (a^{8} v)$
$4 \log (x) - \frac{2}{3} \log (y) - \frac{1}{2} \log (z)$ $\log ⁣ (\frac{x^{4}}{\sqrt[3]{y^{2}} \sqrt{z}})$

How to do these

$\frac{1}{3} \log (x) + 2 \log (y) - 4 \log (z)$

Coefficients become powers: x^(1/3), y² and z⁴.
x^(1/3) is the cube root of x.
Plus terms multiply on top and the minus term divides.

The answer is $\log ⁣ (\frac{\sqrt[3]{x} y^{2}}{z^{4}})$ .

Where students go wrong

Turning a fraction coefficient into division. (1/2)log x is log √x, the log of the square root of x, not log(x/2).

Also called combining logarithms or writing as a single logarithm.

What you can put on this worksheet

Condense into a single logarithm: $\ln (m) + \ln (n)$ → $\ln (m n)$
Fraction coefficients and three terms: $3 \log_{5} (x) - \frac{1}{2} \log_{5} (y) + \log_{5} (z)$ → $\log_{5} ⁣ (\frac{x^{3} z}{\sqrt{y}})$
Numbers, a coefficient over brackets, what cannot condense: Condense: $2 \log_{3} x + 3 \log_{3} y$ → $\log_{3} ⁣ (x^{2} y^{3})$

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.

You pick from 3: condense into a single logarithm, fraction coefficients and three terms and numbers, a coefficient over brackets, what cannot condense. Tick as many as you want and set how many of each, or let it spread them evenly.

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.

Condensing logarithms worksheet for Precalculus

Select your difficulty

Condense into a single logarithm

Condense into a single logarithm

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these condensing logarithms worksheets free?+

Do the condensing logarithms worksheets come with an answer key?+

Are these condensing logarithms worksheets printable?+

What grade are these condensing logarithms worksheets for?+

Can I get a different version of the same condensing logarithms worksheet?+

Is this a free alternative to Kuta Software for condensing logarithms?+

Can I choose how hard the condensing logarithms questions are?+

What kinds of question are on a condensing logarithms worksheet?+

What do students get wrong with condensing logarithms?+