Properties of logarithms worksheet for Precalculus

In Precalculus the log properties are the tool for solving exponential and log equations, so they appear with ln as well as other bases, both expanding one log into several and condensing several into one. Condensing is usually the step that sets up the solve.

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

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Properties of Logarithms Date Period Expand or condense each expression. Expand the logarithm $\log_{7} ⁣ (a^{6} b^{7})$ $\log_{4} ⁣ (\frac{u^{5}}{n^{5}})$ $\log_{5} ⁣ (\frac{a^{6}}{y})$ $\ln ⁣ (m^{6} b^{5})$ $\log_{6} ⁣ (x^{7} n)$ $\ln ⁣ (\frac{u^{7}}{n^{2}})$ $\log_{6} ⁣ (\frac{u^{6}}{y^{2}})$ $\ln ⁣ (p^{6} n)$ $\ln ⁣ (\frac{a^{7}}{v})$ $\ln ⁣ (u^{7} b)$ Condense into a single logarithm $6 \log_{4} (a) - 8 \log_{4} (y)$ $6 \ln (u) - 2 \ln (y)$ $9 \ln (x) + \ln (v)$ $6 \log_{3} (m) + 7 \log_{3} (v)$ $8 \ln (a) + 5 \ln (v)$ $9 \log_{5} (x) + \log_{5} (b)$ $7 \log_{3} (u) - \log_{3} (y)$ $6 \ln (x) - \ln (y)$ $7 \log_{3} (x) - 2 \log_{3} (n)$ $8 \log_{3} (m) + 6 \log_{3} (v)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Properties of Logarithms

Date Period

Expand or condense each expression.

Expand the logarithm
$\log_{7} ⁣ (a^{6} b^{7})$
$\log_{4} ⁣ (\frac{u^{5}}{n^{5}})$
$\log_{5} ⁣ (\frac{a^{6}}{y})$
$\ln ⁣ (m^{6} b^{5})$
$\log_{6} ⁣ (x^{7} n)$
$\ln ⁣ (\frac{u^{7}}{n^{2}})$
$\log_{6} ⁣ (\frac{u^{6}}{y^{2}})$
$\ln ⁣ (p^{6} n)$
$\ln ⁣ (\frac{a^{7}}{v})$
$\ln ⁣ (u^{7} b)$

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



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

Sigma Prep · sigmaprep.io/worksheets

Name

Properties of LogarithmsAnswer keyVersion 2

Date Period

Expand or condense each expression.

Expand the logarithm
$\log_{7} ⁣ (a^{6} b^{7})$ $6 \log_{7} (a) + 7 \log_{7} (b)$
$\log_{4} ⁣ (\frac{u^{5}}{n^{5}})$ $5 \log_{4} (u) - 5 \log_{4} (n)$
$\log_{5} ⁣ (\frac{a^{6}}{y})$ $6 \log_{5} (a) - \log_{5} (y)$
$\ln ⁣ (m^{6} b^{5})$ $6 \ln (m) + 5 \ln (b)$
$\log_{6} ⁣ (x^{7} n)$ $7 \log_{6} (x) + \log_{6} (n)$
$\ln ⁣ (\frac{u^{7}}{n^{2}})$ $7 \ln (u) - 2 \ln (n)$
$\log_{6} ⁣ (\frac{u^{6}}{y^{2}})$ $6 \log_{6} (u) - 2 \log_{6} (y)$
$\ln ⁣ (p^{6} n)$ $6 \ln (p) + \ln (n)$
$\ln ⁣ (\frac{a^{7}}{v})$ $7 \ln (a) - \ln (v)$
$\ln ⁣ (u^{7} b)$ $7 \ln (u) + \ln (b)$

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

How to do these

Condense: $5 \ln (x) - 3 \ln (y)$

Coefficients become powers: ln x⁵ and ln y³.
A difference of logs is the log of a quotient.

The answer is $\ln ⁣ (\frac{x^{5}}{y^{3}})$ .

Where students go wrong

Treating a quotient of logs as the log of a quotient. ln x / ln y is not ln(x/y); the quotient rule only turns a difference of logs into one log.

Also called log rules, the laws of logarithms or expanding and condensing logarithms.

What you can put on this worksheet

Expand into separate logarithms: $\ln (m q)$ → $\ln (m) + \ln (q)$
Condense into a single logarithm: $\ln (m) + \ln (n)$ → $\ln (m n)$
Roots, three factors and numbers: $\log_{3} ⁣ (\frac{x^{4}}{y^{2}})$ → $4 \log_{3} (x) - 2 \log_{3} (y)$
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 only: products, quotients, powers, roots, both directions: Expand:
$\log_{2} ⁣ (\frac{\sqrt[3]{3}}{13})$ → $\frac{1}{3} \log_{2} 3 - \log_{2} 13$

Questions about these worksheets

Yes. Every properties of 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 properties of 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 5: expand into separate logarithms, condense into a single logarithm, roots, three factors and numbers, fraction coefficients and three terms and numbers only: products, quotients, powers, roots, both directions. Tick as many as you want and set how many of each, or let it spread them evenly.

Splitting a sum inside the logarithm. log(x + y) is not log x + log y, and nothing turns it into that. Only products, quotients and powers have rules.

Properties of logarithms worksheet for Precalculus

Select your difficulty

Expand the logarithm

Condense into a single logarithm

Expand the 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 properties of logarithms worksheets free?+

Do the properties of logarithms worksheets come with an answer key?+

Are these properties of logarithms worksheets printable?+

What grade are these properties of logarithms worksheets for?+

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

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

Can I choose how hard the properties of logarithms questions are?+

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

What do students get wrong with properties of logarithms?+