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Math 150, Fall 2008, Benjamin
Aurispa
Chapter 4, Continued
4.3 Laws of Logarithms
1. loga (AB) = loga A + loga B
A
2. loga ( B
) = loga A − loga B
3. loga (Ac ) = C loga A
Examples:
Evaluate the following expressions.
• log12 9 + log12 16
• log2 160 − log2 5
• log2 833
Use Laws of Logarithms to expand the following expressions.
• loga
• ln
x2
yz 3
!
3x2
(x + 1)10
√
3
ln 3r2 s
√
•
ln x2 + 1
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Math 150, Fall 2008, Benjamin
Aurispa
Use Laws of Logarithms to combine the following expressions. (Write as a single logarithm.)
•
1
2
log5 (x2 − 1) − log5 (x − 1)
• 2(log3 x3 + 2 log3 (y + 1) − 3 log3 z)
The following are common errors that students make with logarithms and are NOT true.
• loga (A + B) = loga A + loga B
• loga (AB) = (loga A)(loga B)
A
)=
• loga ( B
loga A
loga B
loga A
loga B
=
A
B
NOT TRUE
NOT TRUE
• (loga A)c = c loga A
•
NOT TRUE
NOT TRUE
NOT TRUE
Sometimes it is necessary to change the base of a logarithm so that you can evaluate it on a calculator or
so that you can write it in terms of the common or natural logarithm.
Change of Base Formula: logb x =
loga x
.
loga b
Examples
• Write log5 2 with common logarithms and evaluate to 4 decimal places.
• Write log6 92 with natural logarithms and evaluate to 4 decimal places.
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Math 150, Fall 2008, Benjamin
Aurispa
4.4 Exponential and Logarithmic Equations
To solve exponential equations,
1. Isolate the exponential expression on one side.
2. Take the logarithm of both sides and simplify using Laws of Logarithms. (If the exponential expression
has base e, use ln.)
3. Solve for the variable.
Examples:
2x = 82x+1
2e12x+4 = 17
32x−1 = 5
e2x + 3ex − 4 = 0
5x+1 = 23x
e3x − 6e2x + 9ex = 0
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Math 150, Fall 2008, Benjamin
Aurispa
To solve logarithmic equations,
1. Isolate the logarithmic term on one side. You may have to use logarithm laws to do this.
2. Write the equation in exponential form. –OR– Exponentiate both sides.
3. Solve for the variable
4. Check for extraneous solutions.
Examples
• ln(2 + x) = 4
• log3 (2 − x) − 3 = 0
• log2 x + log2 (x − 7) = 3
• 2 log x = log 2 + log(3x − 4)
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Math 150, Fall 2008, Benjamin
Aurispa
Applications:
If $6500 is invested in an account at 8% per year compounded semiannually. How long will it take for the
amount in the account to be $7500?
A man invests $6500 in an account that pays 6% interest per year, compounded continuously. How long
will it take for his investment to be $8000?
Suppose the temperature T of an engine after t minutes is given by the equation
T − 20
ln
200
= −0.11t
Solve this equation for T .
Ebbinghaus’ Law of Forgetting states that if a task is learned at a performance level P0 , then after a time
interval t the performance level P satisfies
log P = log P0 − c log(t + 1)
where c is a constant that depends on the type of task and t is measured in months. Solve for P .
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Math 150, Fall 2008, Benjamin
Aurispa
4.5 Modeling with Exponential and Logarithmic Functions
A population that grows exponentially increases according to the model
n(t) = n0 ert
where n(t)=population at time t, n0 =initial size of the population, r=relative rate of growth as a decimal,
and t=time.
Examples
• The number of a certain species of fish is modeled by the function
n(t) = 12e0.012t
where t is measured in years and n(t) is measured in millions.
– What is the initial number of fish?
– What is the relative rate of growth of the fish population (as a percentage)?
– What will the fish population be after 5 years?
– After how many years will the number of fish reach 30 million?
• A culture starts with 1500 bacteria. After 45 minutes the count is doubled. Find a function that
models the number of bacteria n(t) after t hours.
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Math 150, Fall 2008, Benjamin
Aurispa
Exponential functions can also model exponential decay, specifically radioactive decay.
The half-life of a radioactive substance is the time it takes for half the mass to decay.
If m0 is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is
modeled by the function
m(t) = m0 e−rt
Solve for r in terms of the half-life h.
Examples
• The half-life of radium-226 is 1600 years. Suppose we have a 22-mg sample.
– Find a function that models the mass remaining after t years.
– After how long will only 18 mg of the sample remain?
• After 3 days a sample of radon-222 has decayed to 58% of its original amount. What is the half-life of
radon-222?
Some applications of logarithmic functions include the pH scale, the magnitude of earthquakes, and the
decibel scale. You can read more about these in your textbook on pp. 376-379.
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