Algebra 2 Section 8-6 Daily Goals: ·To understand and use the inverse function of the Exponential Function. ·To understand and know how to apply Natural Logarithmic Function properties. ·To be able to solve problems involving the Natural Logarithmic Function and the Exponential Function by applying their inverse property. Natural Logarithms

Warm-Up Use your calculator to estimate, to the nearest thousandth, the following equations of the form y = e x : 1. y = e 3 2. y = e y = e 4. y = e 5 No w use your calculator to evaluate the following expressions, use your approximations from above: 1. ln(answer to 1) 2. ln(answer to 2) 3. ln(answer to 3) 4. ln(answer to 4)

The Natural Logarithmic Function ·In lesson 8-2 you learned that the number e ≈ can be used as a base for exponents. The function y = e x has an inverse, the Natural Logarithmic Function. ·If y = e x, then log e y = x, which is usually written as ln y = x ·The natural log function is the inverse written as y = ln x

Graphical Representation of Inverse

Properties of Natural Logarithm Properties of common logarithms apply to the natural logarithm. ·ln(ab)= ·ln(a/b)= ·ln(a b )=

Proof ·Just for fun lets prove that the ln(ab) = ln(a) + ln(b)

Examples Solving Natural Log Equations ln (4x + 10) 3 = 9 ln x =.1 ln ((x+2)/3)= 12

Examples Write the following as a single logarithm 4 ln(2) – ln(8) 5ln(a) + ln(b) 1/5 ln(6) + 1/5 ln(x)

Examples Solving Exponential Equations 7e 2x = 20 e x+1 = 30

Spacecraft Example ·A spacecraft can attain a stable orbit 300 km above Earth if it reaches a velocity of 7.7 km/s. The formula for a rocket's maximum velocity v in kilometers per second is v = t + c ln R. The booster rocket fires for t seconds and the velocity of the exhaust is c km/s. The ratio of the mass of the rocket filled with fuel to its mass without fuel is R. Suppose R = 25, c = 2.8, and t = 100. Can the spacecraft attain a stable orbit 300 km above Earth?

Investment Example An investment of $100 is now valued at $ The interest rate is 8%, compounded continuously. About how long has the money been invested?

Homework Homework 8-6 ·Page 472 #1-30 (even #’s only) ·Make sure to include your name and class period number at the top of the homework.