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Session #5 ­ Logarithms ­
What is a log? ­ it is the inverse of an exponential equation.
When you take the inverse of an exponential equation with a
base of e, you have a natural logarithm, written as ln.
1. Evaluating Logs
The word log really means ­ What is the exponent? SO a
question like log264 translates into “What is the exponent
on a base of 2 that results in an answer of 64?”
When no base is written, it is understood to be a base of 10.
Ex. Evaluate
1. log 1000
2. log216
3. Find x, logx25 = 2
4. Find x, log432 = x
5. ln e7
6. ln 1
7. ln e
8. log24 + log5125 ­ 2log3243
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1. Log Rules
1. When you are multiplying with the same base you add
the exponents, so if you are multiplying values with the same
base, you add the logs.
logaxy = logax + logay
2. When you are dividing with the same base you subtract
the exponents, so if you are dividing values with the same
base, you subtract the logs.
loga(x/y) = logax ­ logay
When you are raising a power to a power, you multiply the
exponents together, so when you are have the log of
something raised to a power you multiply the power and the
log (in other words, you bring it down front).
logaxy = ylogax
Examples
1. If log 2 = x and log 3 = y, write the following in terms of x
and y.
a. log 9
b. log 18
c. log144
2. Write as a single log: log x = log 3 ­ (2loga + ½ logb)
3. If x = 4x3 , then the logx =
√y5
4. log264 ­ log24
6. 2log4x ­ log4(x + 3) = 1
5. log2x + log2(x ­ 4) = 5
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Exponent Rules and Log Rules
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7. Solve for x: log4(x2 + 3x) ­ log4(x + 5)= 1
8. The expression log 10x+2 ­ log10x is equivalent to
9. If log a = 2 and log b = 3, what is the numerical value of
log (√a)/b3 ?
10. Express as a single natural logarithm (2lnx + lny) - ½ lnz
11. Expand using log rules: ln[(x - 4)(2x + 5)]2
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3. Solving Log Equations
What do you do?
1. When the exponent is a variable and the bases are not
powers of the same base, you will take the logarithm of both
sides of the equation.
2. Isolate what is being raised to the power before taking the
log/ln of both sides.
1. Depreciation (the decline in cash value) on a car can be
determined by the formula V = C(1­r)t, where V is the value
of the car after t years, C is the original cost, and r is the rate
of depreciation. If a car’s cost, when new, is $21,000, the rate
of depreciation is 24%, and the value of the car is now $5,500,
how old is the car to the nearest tenth of a year?
2. The amount A, in milligrams, of a 25­milligram dose of a
drug remaining in the body after t hours is given by the
formula A = 25(.73)t. Find, to the nearest tenth of an hour, how
long it takes for half of the drug dose to be left in the body.
3. Growth of a certain strain of bacteria is modeled by the
equation G = A(2.7)0.584t , where:
G = final number of bacteria
A = initial number of bacteria
t = time (in hours)
In approximately how many hours will 9 bacteria first
increase to 623 bacteria? Round your answer to the nearest
hour.
8 hours ­ at 7 hours it will not have reached 623 yet.
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4. An archaeologist can determine the approximate
age of certain ancient specimens by measuring the
amount of carbon­14, a radioactive substance,
contained in the specimen. The formula used to
determine the age of a specimen is A = A02­t/5760
where A is the amount of carbon­14 that a specimen
contains, is the original amount of carbon­14, t is
time, in years, and 5760 is the half­life of carbon­14. A
specimen that originally contained 201 milligrams of
carbon­14 now contains 83 milligrams of this
substance. What is the age of the specimen, to the
nearest hundred years?
5. Solve for x to the nearest thousandth: 11 + 5e0.24x = 16.578
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5. A hotel finds that its total annual revenue and the
number of rooms occupied daily by guests can best
be modeled by the function R = 3log(n2 + 10n), n > 0
where R is the total annual revenue, in millions of
dollars, and n is the number of rooms occupied daily by
guests. The hotel needs an annual revenue of $12
million to be profitable. Graph the function on the
accompanying grid over the interval 0 < n < 100
Calculate the minimum number of rooms that must be
occupied daily to be profitable.
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6. Solve for x: ln(3x + 2) + ln(x - 1) = ln(3x2 + x - 6)
7. Solve for x in terms of e: 3ln(2x) = 15
8. Show, using the log rules, that ln 2 = ln (½ )-1
9. On January 1, 2010, an initial investment of $10,000 is
made. The money will remain untouched in the account. The
interest is compounded continuously at a rate of 4.1% for the life
of the investment. During what year will the investment reach
$25,000?
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6. Mouthwash manufacturers are constantly testing chemicals
on bacteria that thrive on human saliva. The death of the
bacteria exposed to Antigen 223 can be represented by the
function P(t) = 2,000e-0.37t, where P(t) represents the number of
bacteria from a population of 2000 survivors after t minutes.
a. Determine the number of bacteria surviving after 8 minutes.
b. Determine the number of minutes, to the nearest tenth of a
minute, necessary to kill 90% of the bacteria.
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