1. LEARNING GOALS – LESSON 7.6
7.6.1: Use the number e to write and graph exponential functions
representing real-world situations (compound interest & ½ life.)
7.6.2: Solve equations and problems involving e or natural logarithms.
__________ Interest Formula:
If the interest were to be compounded continuously (n=“∞”) we discover a
horizontal ___________________ at
≈ This number is called _____.
* e is just a __________ and it is _______________
A logarithm with a base of e is called a
_____________ logarithm. Written:
NOT:
f(x) = ln x is the inverse of f(x) = ex.
Example 2: Simplifying Expression with e or ln
Simplify.
A. ln e0.15t
B. e3ln(x +1)
C. ln e2x + ln ex

2. Example 3: Economics Application
Check It Out! Example 2
Simplify.
D. ln e3.2
E. e2lnx
F. ln ex +4y
Continuously Compounded Interest: A = Pert
A – P –
r – t –
Example 3: Economics Application
What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously?
If the total amount for an investment of $500 invested for 50 years and compounded continuously was $10,000 what was the interest rate?
A = Pert
A = Pert

3. Example 4: Science Application
Half-Life of a Substance: N(t) = N0e-kt
N(t) – N0 –
k – t –
Example 4: Science Application
A. Pluonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g?
Step 1 Find the decay constant K for plutonium-239.
N(t) = N0e–kt
Use the natural decay function.
Substitute 1 for N0 ,24,110 for t, and ½ for N(t) because half of the initial quantity will remain.
Simplify and take ln of both sides.
Step 2 Write the decay function and solve for t.
It takes approximately _____________ years to decay.

4. Example 4: Science Application
B. Determine how long it will take for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg.
Step 1 Find the decay constant K for chromium-51.
N(t) = N0e–kt
Step 2 Write the decay function and solve for t.
It takes approximately _____________ days to decay.