Logarithms Essential Question – How is a log function related to an exponential function? You use...

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Transcript of Logarithms Essential Question – How is a log function related to an exponential function? You use...
LogarithmsEssential Question – How is a log function
related to an exponential function?You use log functions to solve exponential problems;
they are inverses of each other.
Math 3 Keeper 27
When will I use this?
• Human memory
• Intensity of sound (decibels)
• Finance
• Richter scale
Evaluating Log Expressions
• We know 22 = 4 and 23 = 8 • But for what value of y does 2y = 6?• Because 22<6<23 you would expect
the answer to be between 2 & 3.• To answer this question exactly,
mathematicians defined logarithms.
• Logarithms are the INVERSE of exponential functions.
Definition of Logarithm to base b
Let b & x be positive numbers, b ≠ 1.
logby = x iff bx = y
• This expression is read “log base b of y”
• The definition tell you that the equations logby = x and bx = y are equivalent.
Log form Exp. form
a) log232= 5
b) log51 = 0
c)log101 = 1
d) Log1/2 2 = 1
25 = 3250 = 1101 = 1(1/2)1 = 2
Example 1: Rewrite the equation in exponential form
Log form Exp. form
e) log39= 2
f) log81 = 0
g)log5(/25)=2
32 = 980 = 152 = 1/25
YOUR TURN!!
Rewriting forms:
To evaluate log3 9 = x ask yourself…
“3 to what power is 9?”
3x = 9 → 32 = 9 so…… log39 = 2
Example 2: Evaluate the expression without a calculator
a)log381
b) log50.04
c) log5125
3x = 81
5x = 0.04
5x = 125
4
2
3
YOUR TURN!!
d) log4256
e) log464
f) log1/4256
g) log2(1/32)
4x = 2564x = 64(1/4)x = 2562x = (1/32)
4
34
5
You should learn the following general forms!!!
•Log b 1 = 0 because b0 = 1
•Log b b = 1 because b1 = b
•Log b bx = x because bx = bx
Natural logarithms
log e x = ln x
• The natural log is the inverse of the natural base, e.
• ln means log base e
Common logarithms
log 10 x = log x
• Understood base 10 if nothing is there.
Common logs and natural logs with a calculator
log10 button
ln button
Example 3: Use a calculator to evaluate the expression. Round
answer to 3 decimal places.
a) log 5
b) ln 0.1
c) log 7
d) ln 0.25
0.6989= 0.700
2.303
0.845
1.386
g(x) = log b x is the inverse of the exponential function f(x) = bx
f(g(x)) = blogbx = x
g(f(x)) = logbbx = x
*Exponential and log functions are inverses and “undo” each other
INVERSE PROPERTIES
a) 10log2 = b) log39x =c) 10logx =
d) log5125x =
2log3(32)x =log332x=2xx
3x
Example 4: Using inverses→ Simplify the expression.
log5(53)x =log553x =
Finding Inverses
Find the inverse of y = log3x
• By definition of logarithm, the inverse is
y=3x
OR write it in exponential form and switch
the x & y! 3y = x → 3x = y
Example 5: Find the inverse of...
a) y = ln (x +1) X = ln (y + 1) Switch the x & y
x = loge(y + 1) Write in log form
ex = y + 1 Write in exp form
ex – 1 = y Solve for y
y = ex – 1 Final Answer
Example 5: Find the inverse of...
b) y = log8x 8y = x Switch x & y
8x = y Solve for y
y = 8x Final Answer
Example 5: Find the inverse of...
c) y = ln (x  3)
d) y = log2/5x
e) Y = ln (x–10)
y = ex + 3
y = (2/5)x
y = ex + 10