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AA3-5 Investigation -- solving exponential models with logarithms cloned 1/19/2021

Last updated about 4 years ago
11 questions
In AA3-1 investigation we saw that exponential functions with (h,k)=(0,0) are written in the form:
-- a is the starting value (f(0))
-- b is the rate of growth or decay (what we are multiplying by each time)
-- x is the number of times something occurs. It is often, but not always, an amount of time.

In AA3-2 we learned the 4 steps to write and solve exponential equations given a function story.

1) Identify the start value.
2) Determine the rate of growth or decay. Remember 0<b<1 is a rate of decay, and b>1 is a rate of growth.
3) Write the exponential equation.
4) Evaluate f(x) for a given value of x.

In AA3-3 we learning how to solve an exponential equation using the definition of a logarithm;

In this investigation we will put all three of these together:
1

Function story: A person invests $100 and it is growing at a rate of 3% per month.

Write the exponential equation in the form:
(Use the equation editor or the ^ will make it an exponent.

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In Investigation AA3-2 we found that after 4 months the investment would be worth $112.55.

Now we are going to answer the question, when will the investment be worth $150?
In other words, solve f(x)=150

Step 1: Write the equation we need to solve.

1

Your equation should be 150 = 100(1.03)x

Step 2: Isolate the grouping symbol.
(1.03)x = ____________________

1

Step 3: Rewrite the exponential equation as a logarithm using the definition:


(1.03)x = 1.5 as a logarithm is:

1

Now solve for x, using either loga in the function menu, or rewriting it using the base change rule:


1

How long will it be before the investment is worth $150?
Round to the nearest month.

1

Try this one:
I have 1400 mosquitoes in my backyard. Each day my bug zapper kills 14% of the mosquitoes.

When will there be 50 mosquitoes in my backyard?

1

When will there be 50 mosquitoes in my backyard?

1

Boots bought a house for $120,000, each year its value goes up by 4%.

When will the house be worth $150,000?

1

When will the house be worth $150,000

1

Write and solve an exponential story problem using a logarithm