 # 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 ! Using the Power Rule we find 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

NOTE: Do not round until you find the final answer! The growth constant k equals approximately 0.0227.

Problem 8:

The half-life of a radioactive substance is 950 years.. Find the constant k rounded to
seven decimal places. Do not use scientific notation! Hint: Half-life means that exactly
one-half 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 one-half of the original amount is left. Therefore, Then The decay constant k equals approximately - 0.0007296.

Problem 9:

The next problem involves carbon-14 dating which is used to determine the age of fossils
and artifacts. The method is based on considering the percentage of a half-life of carbon-
14 of approximately 5715 years. Specifically, the model for carbon-14 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 carbon-14. Estimate the age of the scrolls in 1947.
We know that A, the amount present is 76% of the original amount . Therefore, we  