essential question: what is the relationship between a logarithm and an exponent?
TRANSCRIPT
You’ve ran across a multitude of inverses in mathematics so far...◦ Additive Inverses: 3 & -3◦ Multiplicative Inverses: 2 & ½ ◦ Inverse of powers: x4 & or x¼
◦ But what do you do when the exponent is unknown? For example, how would you solve 3x = 28, other than guess & check?
◦ Welcome to logs…
4 x
Logs◦ There are three types of commonly used logs
Common logarithms (base 10) Natural logarithms (base e) Binary logarithms (base 2)
◦ We’re only going to concentrate on the first two types of logarithms, the 3rd is used primarily in computer science.
◦ Want to take a guess as to why I used the words “base” above?
The logarithm to the base b of a positive number y is defined as follows:◦ If y = bx, then logby = x
◦ All logs can be thought of as a way to solve for an unknown exponent logbase answer = exponent
log10
10 2x =
x
10 2x =
Example: Write “25 = 52” in logarithmic form◦ Remember: logbase answer = exponent
◦ log5 25 = 2
Your turn: Write the following exponential equations in logarithmic form.◦ 729 = 36
◦ (1/2)3 = 1/8
◦ 100 = 1
log3 729 = 6
log1/2 1/8 = 3
log10 1 = 0
Example: Write “log8 16 = x” in exponential form◦ Remember: logbase answer = exponent◦ 8x = 16
Your turn: Write the following exponential equations in logarithmic form.◦ log64 1/32 = x
◦ log9 27 = x
◦ log10 100 = x
64x = 1/32
9x = 27
10x = 100
Assignment◦ Page 450◦ Problems 7 – 25 & 53 – 61 (odd problems)
For questions 15 – 25, pretend they “= x” We’ll deal with how to solve them Tuesday
Common logarithms◦ Scientific/graphing calculators have the common
and natural logarithmic tables built in.◦ On our calculators, the “log” button is next to
the carat (^) key.◦ To find log10 29, simply type “log 29”, and you
will be returned the answer 1.4624. That means, 101.4624 = 29
◦ Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places
Solving Logarithmic Equations (w/o calculator)◦ log216 = x
Can be rewritten as 2x = 16. Because 24 = 16, x = 4
◦ log5(-25) = x Rewritten as 5x = -25, which isn’t possible. Undefined
◦ log5x = 3 Can be rewritten as 53 = x, so x = 125
Example: Solve for xlog8 16 = x
Your turn: Solve for x◦ log64 1/32 = x
◦ log9 27 = x
◦ log10 100 = x
-0.8333
1.5
2
8
log16log 16 1.3333
log 8