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SOLVING EXPONENTIAL EQUATIONS
Problem 3:
Solve
Method 1:
Let's use the natural logarithm in the solution process!
Notice the parentheses around (x  1)!!!
Method 2:
Please be aware that we can only use this method because the numbers
and have the same base when written in
exponential form . That is,
and .
Therefore, let's rewrite as
and
Since the two expressions are obviously equal, and the bases are also
obviously equal, then the two exponents also MUST be equal to each other.
Therefore,.
and
Problem 4:
Solve: not
using logarithms!
Let's solve this exponential equation using the fact that 9 and 27
have the same base!
That is,
Problem 5:
Solve . Round to 3 decimal places .
Let's use the common logarithm in the solution process!
Next, we distribute the logarithmic expressions
and collect the expressions containing the variable on one side
this allows us to factor out the variable and isolate it as follows
Problem 6:
How many years will it take for an initial investment of $10,000 to grow
to $25,000?
Assume a rate of interest of 2.5% compounded continuously. Round your
answer to a
whole number. Use the formula , where P
is the initial investment, A is the
accumulated amount, t is the time in years and r is the interest rate in
decimals.
NOTE: Do not round until you find the final answer!
takes approximately 37 years for $10,000 to grow to
$25,000 at a rate of
interest of 2.5%.
Problem 7:
The number of bacteria A in a certain culture is given by the growth
model
. Find the growth constant k knowing
that A = 280 when t = 5. Round
your answer to four decimal places .
NOTE: Do not round until you find the final answer!
The growth constant k equals approximately 0.0227.
Problem 8:
The halflife of a radioactive substance is 950 years.. Find the constant
k rounded to
seven decimal places. Do not use scientific notation! Hint: Halflife means that
exactly
onehalf of the original amount or size of the substance is left after a certain
number of
years of growth/decay. Use the Exponential Growth/Decay Model
, where
is the original amount, A is the
accumulated amount, t is the time in years and k is the
growth constant.
We know that after 950 years onehalf of the original amount
is left. Therefore,
Then
The decay constant k equals approximately  0.0007296.
Problem 9:
The next problem involves carbon14 dating which is used to determine the age of
fossils
and artifacts. The method is based on considering the percentage of a halflife
of carbon
14 of approximately 5715 years. Specifically, the model for carbon14 is
.
In 1947, an Arab Bedouin herdsman found earthenware jars containing what are
known
as the Dead Sea scrolls. Analysis at that time indicated that the scroll
wrappings
contained 76% of their original carbon14. Estimate the age of the scrolls in
1947.
Round your answer to a whole number.
We know that A, the amount present is 76% of the original amount
. Therefore, we
can use the model to write
The Dead Sea Scrolls were approximately 2268 years old in 1947.
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