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Exponents and radicals practice

Exponential equations without logarithms worksheets

Solve nine types of exponential equations by rewriting to a common base. Choose 12 or 14 questions with exact fraction answers and worked keys.

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Rewrite both sides using the same base. Equate their exponents and solve for the variable. Give exact answers without a calculator.

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Rewrite both sides with one base

Choose any of nine equation forms and 12 or 14 questions. Practice matching bases, combining products, subtracting exponents in a quotient, and rewriting square, cube and reciprocal bases.

All questions reduce to a linear equation with an exact answer. The key shows the common-base rewrite and the exponent equation. Keep fractions instead of rounding; no logarithms or calculator are needed.

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

Rewrite related integer and reciprocal bases, combine exponents, and solve the resulting linear equation exactly.

Worked example

Solve (1/2)^(x + 1) times 4^(2x) = 1/4.

Rewriting with base 2 gives -x - 1 + 4x = -2. Therefore 3x = -1 and x = -1/3.

The reciprocal base contributes a negative exponent, while base 4 contributes twice its exponent. Once both sides have the same positive base greater than one, their exponents must be equal.

Teaching notes

Make the practice count

  • Look for a common base before doing any algebra.
  • A quotient of powers means subtracting the entire denominator exponent.
  • Rewrite a reciprocal as a negative power.
  • Keep fractional solutions exact, even when the original exponential values are irrational.

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Customize the practice

Choose your options, update the preview, then download the exact same pages.

Choose equation forms and 12 or 14 questions. Rewrite to a common base and solve exactly, without a calculator.

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Configured PDF · Exponential Equations Without Logarithms · US Letter

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Math SheetNameScore/ 14TeacherDateExponential Equations Without LogarithmsRewrite each equation with a common base, equate the exponents, and solve exactly.1.Rewrite using base 7.7-2·q7-4·q+374·qDomain: real variablesqR2.Rewrite using base 4.43·n+344·nDomain: real variablesnR3.Rewrite using base 7.17n+2·3432·n1343Domain: real variablesnR4.Rewrite using base 7.494·m-2·7m343Domain: real variablesmR5.Rewrite using base 3.3-2·n+2·273-4·nDomain: real variablesnR6.Rewrite using base 7.7-z·7-4·z1343Domain: real variableszR7.Rewrite using base 6.62·n+1·6n62·nDomain: real variablesnR8.Rewrite using base 10.104·y+31000Domain: real variablesyRmath-sheet.comPage 1 / 2
Math SheetNameScore/ 14TeacherDateExponential Equations Without LogarithmsRewrite each equation with a common base, equate the exponents, and solve exactly.9.Rewrite using base 4.4-4·n·43·n16Domain: real variablesnR10.Rewrite using base 8.8-z+2·8-z8-zDomain: real variableszR11.Rewrite using base 2.2m·2m14Domain: real variablesmR12.Rewrite using base 5.152·y+1·253·y125Domain: real variablesyR13.Rewrite using base 5.5-2·q+125Domain: real variablesqR14.Rewrite using base 7.7-q+373·qDomain: real variablesqRmath-sheet.comPage 2 / 2

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