5.4 B Notes

5.4 "Logarithmic Functions"
32 = 2x
3 = 27x 100 = 5x How do we solve for the variable?
y = loga x if and only if x = ay
Logarithmic Form:
Exponential Form:
a y = x
log a x = y
(ex ) Change to exponential form:
("A logarithm is an exponent")
a) log 3 81 = 4 b) log 7 7 = 1 c) log 14 1 = 0 d) log 1/2 32 = 5
(ex) Change to logarithmic form.
a) 2 3 = 8 b) 4 0 = 1 c) 12 1 = 12 d) (1/4) 1 = 4
"The logarithm of 8 with base 2 is 3"
exponent
(try) Change to exponential form.
a) log 3 81 = 4 b) log 4 (1/256) = 4 c) log v w = q d) log 6 (2x 1) = 3
e) log 4 P = 5 x f) log a 343 = 3/4
(try) Change to logarithmic form.
a) 3 5 = 243
b) 3 4 = 1/81
c) c p = d
d) 7 x = 100p
e) 3 2x = p/f
f) (0.9) t = 1/2
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Solve for t using logarithms with base a
(ex) 3a4t = 10
(ex) F = D + Bat Common Logarithm
(ex) L = Mat/n P
Natural Logarithm
log10 x log x
loge x ln x
(ex) Change to logarithmic form.
a) 102 = x
b) 10y+3 = z
c) e2 = x
d) ey+3 = z
(try) Change to logarithmic form.
a) 104 = 10,000 b) 102 = 0.01 c) 10x = 38z d) e4 = D e) e0.1t = x+2
(try) Change to exponential form.
a) log x = 8 b) log x = y2 c) ln x = 1/2 d) ln z = 7+x e) ln(t5) =1.2
(ex) Find the number:
a) log 10 100 b) log 2 (1/32)
d) log 7 1
e) log 3 (2)
g) log 5 0.2
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h) log 1/5 125
c) log 9 3
f) log 4 64
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Some General Properties:
1. log a 1 = 0 because a0 = 1
(ex) log 7 1 = 0
2. log a a = 1 because a1 = a
(ex) log 2 2 = 1
3. log a ax = x because ax = ax
(ex) log 10 104 = 4 (ex) log39=log332=2
4. because (look at example) (ex) (ex)
(ex)
(ex) Find the number.
a) 10 log 7 b) log 106 c) log 100,000
d) log 0.001
e) e ln 8
f) ln e g) e 1 + ln 5
Solve the equation.
(ex) log 3 (x + 4) = log 3 (1 x )
(ex) log 7 (x 5) = log 7 (6x)
(ex) ln x2 = ln (12 x)
(ex) log 2 (x 5) = 4
(ex) log 4 x = (ex) log x2 = 4
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(ex) e ln x = 0.2
(ex) e x ln 2 = 0.25
Approximate x to three significant figures.
a) log x = 1.8965
b) log x = 4.9680
c) log x = 2.2118
d) ln x = 3.7
e) ln x = 0.95
f) ln x = 5
Using your calculator:
10x = 251.19
10x = 900
ex = 90
Doubling Time. A population is growing continuously at a rate of 4% per year.
What's its doubling time? (growth formula: q = q0ert )
Bismuth Decay. The radioactive bismuth isotope 210Bi disintegrates according to Q = k(2)t/5 , where k is a constant and t is the time in days. Express t in terms of Q and k.
Population Growth. The population N(t) (in millions) of India t years after 1985 may be approximated by the formula N(t) = 766e0.0182t . When will the population reach 1.5 billion?
Continuously Compounded Interest. If interest is compounded continuously at the rate of 6% per year, approximate the number of years it will take an initial deposit of $6000 to grow to $25,000.
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