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General functions practice

Graphing logarithmic functions worksheets

Find exact domains and all-real ranges, then graph eight logarithmic functions with vertical asymptotes on fixed square grids. Includes affine arguments, fractional boundaries and separate answer keys.

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Set the logarithm argument greater than zero to find the domain and vertical asymptote. Plot exact points where the argument is 1 and 1/base, then draw a smooth increasing curve on the domain side of the asymptote.

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Find the boundary. Stay on the domain side.

Each worksheet has eight logarithmic functions. For every task, find the exact domain and range, then graph the curve on a square grid from −10 to 10.

For y = log_b(mx + c) + k, solve mx + c > 0. The exact boundary x = −c/m is a vertical asymptote and is not part of the domain. The range is all real numbers.

Plot a point where the argument is 1, which gives y = k. A second exact point where the argument is 1/b lies one unit lower. Draw a smooth increasing curve that approaches the asymptote without touching it.

Learning objective

What this activity builds

Connect a logarithmic equation to its strict domain, vertical asymptote and increasing curve.

Worked example

Find the domain and range, then graph y = log₃(2x − 4) + 1.

Domain x > 2; range all real numbers; vertical asymptote x = 2.

The argument requires 2x − 4 > 0. At x = 5/2 the argument is 1, so y = 1. At x = 13/6 the argument is 1/3, so y = 0.

Teaching notes

Make the practice count

  • Solve mx + c > 0 exactly before graphing.
  • Draw x = −c/m as a vertical asymptote.
  • Use the argument value 1 to plot a point at height k.
  • Use the argument value 1/base to plot a second point one unit lower.
  • Keep every curve on the side where mx + c is positive.
  • Remember that a logarithmic function has all real outputs.

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Math SheetNameScore/ 24TeacherDateGraphing Logarithmic FunctionsFind the domain and range, then graph each logarithmic function.1. y =log5x-1Domain: ____________________Range: _____________________-10-5510-10-50510xy2. y =log10x+5-2Domain: ____________________Range: _____________________-10-5510-10-50510xy3. y =log3x+3+3Domain: ____________________Range: _____________________-10-5510-10-50510xy4. y =log4x-4+1Domain: ____________________Range: _____________________-10-5510-10-50510xymath-sheet.comPage 1 / 2
Math SheetNameScore/ 24TeacherDateGraphing Logarithmic FunctionsFind the domain and range, then graph each logarithmic function.5. y =log83·x-1+3Domain: ____________________Range: _____________________-10-5510-10-50510xy6. y =log74·x+5-2Domain: ____________________Range: _____________________-10-5510-10-50510xy7. y =log84·x-5+5Domain: ____________________Range: _____________________-10-5510-10-50510xy8. y =log62·x-1+2Domain: ____________________Range: _____________________-10-5510-10-50510xymath-sheet.comPage 2 / 2

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