8 6 Ticket Out Use natural logarithms to

8 -6 Ticket Out • Use natural logarithms to solve e– 6 x = 3. 1. x ≈ -0. 189 -6 x = ln 3. 1 • Solve ln 2 + ln x = 5. 2 x = e 5 x ≈ 74. 207 ln (2 x) = 5 • Write 3 ln a – ½ (ln b + ln c 2) as a single natural logarithm. ON BOARD • Simplify ln e– 4. -4

8. 4 – 8. 6 Quiz Review Problems • • • p. 449 #11, 12, 14, 15 p. 456 #1, 3, 5, 6, 9, 10, 33, 34, 35, 40 p. 457 #42, 44 p. 464 #5, 6 p. 465 #17, 18, 23, 24, 26

What are logarithms?

What is a logarithm anyway? • A logarithm is just a way to represent finding an exponent. • A logarithm is the power to which a number must be raised in order to get some other number. Example: log 100 = 2 because 102 = 100. What if the logarithm is not a common log? (Not base 10) The base is the number being raised to a power. There are logarithms using different bases. If you wanted, you could use 2 as a base. Example: log 28 = 3 because 23 = 8.

An Important Concept of Logarithms You can never take the log of a negative number or 0. the s i m h arit “A log o which a t power st be raised r mu e me b o m s u t n e r to g e d r o r. ” e in b m u n other Why? Can you raise any number to any real number to change its sign? NO! Can you raise any number to any real number to get 0? NO! When solving logarithmic equations, always check for extraneous solutions!

Natural Logs ln • ln is the notation for a natural logarithm. • The base of a natural log is always e. • ln x would give us the power that e would have to be raised to equal x. – Ex: ln(7. 38…) = 2 because e 2 = 7. 38… • ln(e) = 1 because e 1 = e. • ln(1) = 0 because e 0 = 1. • The inverse of ln x is ex. (We will prove this later!)

Natural Logs e • e is a constant that is an irrational number (just like pi!) • e is approximately 2. 718. • The inverse of ex is ln x.

The inverse of ln x is ex? Recall the definition of an inverse function: – If you plug x into the original function you get y. Then, if you plug y into the inverse function you get x.

The inverse of ln x is x e? RECALL EXAMPLES LIKE THIS: f-1(f(5)) = 5 OR f-1(f(19984334)) = 19984334 So, we can say that f-1(f(x)) = x

The inverse of ln x is ex? In your calculator, try the following examples: So, we can say that ln(e 9) x) = x & elnx = x ln(e ln 5 e ln(e 58430) By definition, [f-1(f(x)) = x] ln and e are inverses of each other.

When will we ever see logs in real life? ? !!

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