Properties of Logarithms

20 questions

12th - 12th Grade

Science

Properties of Logarithms is a Grade 12 worksheet with 20 multiple-choice questions focused on transforming logarithmic expressions. Students practise condensing several logarithms into a single expression, expanding logarithms into sums or differences, and simplifying expressions by applying logarithm properties. One item, for example, asks students to write log 6 − log 3 + 2 log 7 as a single logarithm and simplify it to log 98. The worksheet gives students repeated practice recognizing which logarithm property fits each expression and selecting the correctly transformed result from multiple-choice options. It is useful for independent practice, guided review, or a quick check of students’ fluency with logarithmic manipulation before more advanced algebraic work. This free printable worksheet includes a complete answer key for efficient checking and feedback.

Worksheets

Properties of Logarithms

Total questions: 20

Name

Class

Date

Condense the Logarithm $5\log_{ }a\ -\ 25\log_{ }b$

$\log\ \left(a^5+b^{25}\right)$

$\log\ \left(a^5-b^{25}\right)$

$\log\ \left(ab\right)^{25}$

$\log_{ }\ \left(\frac{a^5}{b^{25}}\right)$

Condense $3\log_{ }x+4\log_{ }y\ +\log_{ }z$

$\log_{ }\ x^3y^4z$

$12\log_{ }\ xyz$

$\log_{ }\ 3x4yz$

logx³y³z³

Simplify: $\log_4\left(x+4\right)-\log_4\left(x-5\right)$

$\log_49$

$\log_4\left(2x-1\right)$

$\log_4\left(x^2-x-20\right)$

$\log_4\left(\frac{x+4}{x-5}\right)$

Simplify: $2\log_3\left(11x\right)$

$\log_3\left(22x\right)$

$\log_3\left(121x\right)$

$\log_3\left(121x^2\right)$

$\log_3\left(11x^2\right)$

Simplify: $\frac{1}{4}\log_516+3\log_5x$

$\log_5\left(4x^3\right)$

$\log_5\left(2x^3\right)$

$\log_5\left(6x\right)$

$\log_5\left(\frac{2}{x^3}\right)$

Simplify: $\frac{1}{4}\log_281+\frac{1}{2}\log_249$

$\log_221$

$\log_210$

$\log_2\left(\frac{3}{7}\right)$

$\log_244.75$

Simplify: $\frac{1}{2}\log_964+\log_9x$

$\log_9\left(32x\right)$

$\log_9\left(8x\right)$

$\log_9\left(\frac{8}{x}\right)$

$\log_98x$

Write the expression as a single logarithm. Then simplify if possible.
log 6 - log 3 + 2 log 7

log 98

log 78

log 56

log 45

Expand using the properties of Logaritms $\log_{ }\ \frac{x^3}{y^4z}$

$\log_{ }x+4\log_{ }y\ +\log_{ }z$

$3\log_{ }x-4\log_{ }y\ -\log_{ }z$

$3\log_{ }x+4\log_{ }y\ +\log_{ }z$

$3\log_{ }x-4\log_{ }y\ +\log_{ }z$

10.

Expand the logarithm.
$\log\frac{x}{y^6}$

$\log x+6\log y$

$\log x-6\log y$

$\log x+\log6y$

$\log x-\log6y$

11.

Expand . $\log_6\left(\frac{5x^3}{y}\right)$

$\log_65x^3-\log_6y$

$\log_65+\log_6x^3-\log_6y$

$\log_65+3\log_6x-\log_6y$

$\log_65+3\log_6x+\log_6y$

12.

Expand: $\log_6\left(\frac{5}{4y}\right)$

$\log_65\ +\ \log_64\ +\ \log_6y$

$\log_65\ -\ \log_64\ +\ \log_6y$

$\log_65\ -\ \log_64\ -\ \log_6y$

$\log_65\ -\ \log_64y$

13.

Condense: $\log\left(5x+2\right)\ -\ \log3\ -\ \log x$

$\log\left(\frac{3x}{5x\ +\ 2}\right)$

$\log\left(\frac{5x\ +\ 2}{3x}\right)$

$\log\left(3x\left(5x+2\right)\right)$

$\log\left(\frac{5x\ +\ 2}{3\ -\ x}\right)$

14.

Expand: $\log_4\left(3x^2\right)$

$2\log_43x$

$2\log_43\ +\ 2\log_4x$

$\log_43\ +\ 2\log_4x$

$2\log_43\ +\ \log_4x$

15.

Condense: $\frac{1}{2}\log_3x\ +\ 5\log_3y$

$\log_3\left(\sqrt{x}y^5\right)$

$\log_3\left(5\sqrt{x}y\right)$

$\log_3\left(xy\right)^{\frac{5}{2}}$

16.

Condense: $3\ln x\ +\ 8\ln y$

$\ln\left(\frac{x^3}{y^8}\right)$

$\ln\left(x^3y^8\right)$

$24\ln\left(xy\right)$

$11\ln\left(xy\right)$

17.

Condense: $3\log_bx\ -\ \log_by\ -\ 5\log_bz$

$\log_b\ \frac{x}{yz^5}$

$\log_b\ \frac{x^3}{yz^{ }}$

$\log\frac{x}{yz^5}$

$\log_b\ \frac{x^3}{yz^5}$

18.

Condense

logxyz³

log(xy/z³)

logx+logy+logz³

logx³y³z³

19.

Condense

log₈(a¹⁸/b³)

log₈(a¹⁸b³)

log₈(a/b³)

15log₈(a/b³)

20.

Change to Exponential Form:
log₆36 = 2

2⁶=36

6²=36

36²=6

36⁶=2

Answer Key

Properties of Logarithms

Total questions: 20

$\log_{ }\ \left(\frac{a^5}{b^{25}}\right)$

$\log_{ }\ x^3y^4z$

$\log_4\left(\frac{x+4}{x-5}\right)$

$\log_3\left(121x^2\right)$

$\log_5\left(2x^3\right)$

$\log_221$

$\log_98x$

a) log 98

$3\log_{ }x-4\log_{ }y\ -\log_{ }z$

10.

$\log x-6\log y$

11.

$\log_65+3\log_6x-\log_6y$

12.

$\log_65\ -\ \log_64\ -\ \log_6y$

13.

$\log\left(\frac{5x\ +\ 2}{3x}\right)$

14.

$\log_43\ +\ 2\log_4x$

15.

$\log_3\left(\sqrt{x}y^5\right)$

16.

$\ln\left(x^3y^8\right)$

17.

$\log_b\ \frac{x^3}{yz^5}$

18.

a) logxyz³

19.

a) log₈(a¹⁸/b³)

20.

b) 6²=36

140 %

FAQs

What does the Properties of Logarithms worksheet cover?

This worksheet covers condensing, expanding, and simplifying logarithmic expressions through 20 multiple-choice questions. Students must identify the appropriate logarithm property, transform the expression, and choose the correct result, including combining terms such as log 6 − log 3 + 2 log 7 into the single logarithm log 98.

How can I use this worksheet to teach properties of logarithms?

Introduce the worksheet by reviewing how logarithm properties transform products, quotients, and powers, then model one condensing and one expanding example. Have students explain which property they used before selecting an answer, and use the multiple-choice options to discuss why incorrect transformations do not work. Finish with the simplifying items as an independent check of whether students can apply the properties without step-by-step prompting.

What mistakes should I watch for when students complete this logarithms worksheet?

Watch for students reversing the direction of a property when expanding or condensing, especially when handling subtraction and coefficients. Students may also combine terms incorrectly, treat a coefficient as a separate logarithm instead of a power, or stop after condensing without simplifying when simplification is possible. Ask them to name the property used and verify the transformed expression before choosing an option.

How do I assign and grade the Properties of Logarithms worksheet?

The worksheet is available as a printable PDF and as a digital quiz on Wayground, and it includes a complete answer key for checking the 20 multiple-choice responses. Printable submissions can be graded by scanning student work with the Wayground for Teachers app, while the digital version can be assigned through Wayground for online completion and results.

How do the expanding and condensing questions differ on this worksheet?

The expanding questions ask students to rewrite a logarithmic expression into multiple terms, while the condensing questions ask them to combine multiple logarithms into one expression. Together with the simplifying items, these two directions of practice help students recognize logarithm properties whether they are breaking an expression apart or rebuilding it.

Where can I find more worksheets like this on Wayground?

Wayground offers a broad collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to support essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

Properties of Logarithms

Properties of Logarithms

Total questions: 20

Properties of Logarithms

Total questions: 20

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