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Exponents and radicals practice

Properties of logarithms worksheets

Expand and condense logarithms with 16 selectable product, quotient and power forms. Practice natural logs, base-ten logs and fractional powers with exact symbolic answers.

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Expand each selected logarithm or condense the expression to one logarithm. Assume all variables are positive and keep every term on the same base.

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Expand and condense with the same base

The default sheet has six expansion and six condensation questions. Select any of the 16 forms, or choose twelve questions in just one direction. Forms include products, quotients, outer powers and exact fractional exponents.

All variables are positive. Natural logarithms use ln; common logarithms display base 10 explicitly. Keep one base throughout each expression. The key shows an equivalent structural expansion or a single logarithm, without requiring decimal evaluation.

The preview and downloaded PDF use the same resolved questions. Answers follow on separate pages; choose the student-only edition when handing out the exercise.

Learning objective

What this activity builds

Use product, quotient and power laws to rewrite equivalent logarithmic expressions under positive-variable assumptions.

Worked example

Expand ln(x^3 / y^2), with x and y positive.

3 ln(x) - 2 ln(y).

A quotient becomes a difference of logarithms. Each exponent becomes a coefficient, and every term retains the natural-log base.

Teaching notes

Make the practice count

  • Start with two-factor products before introducing quotient signs.
  • Apply an outer power to every factor in the argument.
  • Treat a reciprocal power as an exact fractional coefficient.
  • Accept equivalent factored expressions when condensing, while checking the common base.

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Configured PDF · Properties of Logarithms · US Letter

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Math SheetNameScore/ 12TeacherDateProperties of LogarithmsExpand or condense each logarithmic expression.1.Expand using logarithm properties.ln75·32.Expand using logarithm properties.log10y·z·s12Domain: variables are positivey0z0s03.Expand using logarithm properties.log73674.Expand using logarithm properties.lna3u6Domain: variables are positivea0u05.Expand using logarithm properties.log1096·65·76.Expand using logarithm properties.log6r·m5·y3Domain: variables are positiver0m0y0math-sheet.comPage 1 / 2
Math SheetNameScore/ 12TeacherDateProperties of LogarithmsExpand or condense each logarithmic expression.7.Condense to one logarithm.5·ln2+2·ln8+5·ln98.Condense to one logarithm.6·log10r+6·log10pDomain: variables are positiver0p09.Condense to one logarithm.2·log95+log94+log9210.Condense to one logarithm.lnz+5·lncDomain: variables are positivez0c011.Condense to one logarithm.log103+log106+log10412.Condense to one logarithm.log15q-2·log15aDomain: variables are positiveq0a0math-sheet.comPage 2 / 2

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