# 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 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.

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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