Use this page with Slide 4: Exploring Logarithmic Functions of Fitting Wealth-v-Health and Exploring Logarithmic Models (Desmos).

  • The blue curve is the graph of 𝒽(𝑥) = 1 log2 𝑥 + 0. Its constants will remain set at 𝑎 = 1, 𝑏 = 2, and 𝑐 = 0.

  • You can modify the red curve 𝑔(𝑥) (which is hiding behind h!) by changing its coefficients: 𝑎, 𝑏, and 𝑐.

Base 𝑏

Keep c at 0 and a at 1. Change the value of b as indicated on each grid below.

1 Sketch each graph and label the coordinates where 𝑥 = 1, 𝑦 = 1, 𝑦 = 2 and 𝑦 = 3.

𝑏 = 3

𝑏 = 5

𝑏 = 10

2 How does the value of 𝑏 impact the shape of a logarithmic function?

3 What connections can you draw between the value of 𝑏 and exponents?

Vertical Shift 𝑐

Set a to 1 and b to 2. Change the value of c as indicated on each grid below.

4 Sketch each graph and label the coordinate where 𝑥 = 1.

𝑐 = - 10

𝑐 = 0

𝑐 = 10

5 How does the value of 𝑐 impact the shape of a logarithmic function?

6 Why does 𝑦 = 𝑐 when 𝑥 = 0?

Logarithmic Coefficient 𝑎

Set c to 0 and b to 10, then zoom out out so you can see as far as 𝑥 = 1,000.

  • Change 𝒽(𝑥) to 𝒽(𝑥) = 1 log10 (𝑥) + 0 so that the blue curve lands on top of the red curve.

  • Now you’re ready to change the value of a as indicated on each grid below.

7 In each graph, label the coordinates where 𝑥 = 10 and 𝑥 = 100 and 𝑥 = 1000.

𝑎 = - 2

𝑎 = 0

𝑎 = 2

8 What is the value of 𝑥 when log10(𝑥) = 6? What about when 2 log10(𝑥) = 6? When 3 log10(𝑥) = 6?

★ How are 𝑎 and 𝑏 related?

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