Montana Math MT.PLUS.ALG.EXP.3

MT.PLUS.ALG.EXP.3 is a cluster heading. These are the individual standards under it.

• MT.PLUS.ALG.EXP.3.a

  Expressing exponential functions in the form f(x) = a · b^x and f(x) = Pe^(r·t), and

• MT.PLUS.ALG.EXP.3.b

  Expressing logarithmic functions in base 10 and base e.

01

What Students Need to Do

Students write the same growth or decay rule with an ordinary base and with base e. They use base-10 and natural logs, and rewrite a base-10 log using natural logs.

02

What Mastery Looks Like

Given 300(1.04)^t, a student writes 300e^((ln 1.04)t) and identifies ln 1.04 as the rate in that form. They rewrite log₁₀(100) as ln(100)/ln(10) and get 2.

03

Common Misconceptions

Students may use 0.04 instead of ln(1.04) when changing 1.04^t to base e. They may treat log₁₀(x) and ln(x) as equal, or put ln(10) in the numerator.

04

How to Assess It

Exit ticket: Rewrite 400(1.05)^t with base e and label the new rate. Then rewrite log₁₀(50) using ln.

1. 01

   Use calculators to fill a table for 3(2)^x and 3e^((ln 2)x) at x = 0, 1, and 2.

2. 02

   Ask students to explain in writing why log₁₀(100) and ln(100) give different numbers.

3. 03

   Play a matching-card game with pairs such as 2(5)^x and 2e^((ln 5)x), or log₁₀(1000) and ln(1000)/ln(10).

4. 04

   Give students a $200 account growing 5% yearly and ask them to write its balance using bases 1.05 and e.

• MT.PLUS.ALG.EXP.4

  Graph logarithmic and exponential functions with and without the use of technology by identifying intercepts, asymptotes, and end behavior.

• MT.PLUS.ALG.EXP.1

  Understand logarithmic functions as the inverse of exponential functions.