8-6 Ticket Out• Use natural logarithms to solve e–6x = 3.1.

-6x = ln 3.1• Solve ln 2 + ln x = 5.

ln (2x) = 5• Write 3 ln a – ½ (ln b + ln c2) as a single natural

logarithm.ON BOARD

• Simplify ln e–4.-4

x ≈ -0.189

2x = e5 x ≈ 74.207

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?

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:log28 = 3

because 23 = 8.

•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 = 2because 102 = 100.

An Important Concept of Logarithms

You can never take the log of a negative number or 0. Why?

“A logarithm is the

power to which a

number must be raised

in order to get some

other number.”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 Logsln

• 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 e2 = 7.38…

• ln(e) = 1 because e1 = e.• ln(1) = 0 because e0 = 1.• The inverse of ln x is ex. (We will prove this later!)

Natural Logse

• 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.

RECALL EXAMPLES LIKE THIS:f-1(f(5)) = 5

ORf-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:

ln(e9)eln5

ln(e58430)

So, we can say thatln(ex) = x & elnx = x

By definition, [f-1(f(x)) = x]

ln and e are inverses of each other.

The inverse of ln x is ex?

When will we ever see logs in real

life??!!

• http://www.media4math.com/MathLabs/MathLab8.html

• http://www.liveleak.com/view?i=043_1189839647