Exponential Functions - Pre-Calculus

The -intercept of a graph is the point on the graph where the -value is .

Thus, to find the -intercept, substitute and solve for .

Thus, we get:

Exponential decay describes a function that decreases by a factor every time increases by .

These can be recognizable by those functions with a base which is between and .

The general equation for exponential decay is,

where the base is represented by and .

Thus, we are looking for a fractional base.

The only function that has a fractional base is,

.

The -intercept of a function is where . Thus, we are looking for the -value which makes .

If we try to solve this equation for we get an error.

To bring the exponent down we will need to take the natural log of both sides.

Since the natural log of zero does not exist, there is no exponent which makes this equation true.

Thus, there is no -intercept for this function.

Exponential graphs can either decay or grow. This is based on the value of the base of the exponent. If the base is greater than , the graph will be growth. And, if the base is less than , then the graph will be decay. In this situation, our base is . Since this is greater than , we have a growth graph. Then, to determine the y-intercept we substitute . Thus, we get:

for the y-intercept.

Exponential graphs can either decay or grow. This is based on the value of the base of the exponent. If the base is greater than , the graph will be growth. And, if the base is less than , then the graph will be decay. In this situation, our base is . Since this is less than , we have a decay graph. Then, to determine the y-intercept we substitute . Thus, we get:

for the y-intercept.

Note that the negative sign in this function comes outside of the parentheses. This should show you that the bigger the number in parentheses, the lower the curve of the graph will go. Since this is an exponential function, the larger that the x value gets, then, the "more negative" this graph will go. The graph closest to zero on the left-hand side - where x is negative - and then shoots down and to the right rapidly when x gets larger is the correct graph.

First, simplify the left side of the equation using the additive rule for exponents.

.

Our equation now becomes:

Equating we set the exponents equal to eachother and solve.

Thus,

First, use the additive property of exponents to simplify the right side of the equation.

.

Thus,

.

Now, take the natural log of both sides

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Use the multiplicative property of logarithms to expand the left side to get

Now, apply the logarithms to the exponents

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Rearrange to get the x-terms on one side

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Finally, divide the 2 on both sides

.

First, let's begin by simplifying the left hand side.

becomes and becomes . Remember that , and the in that expression can come out to the front, as in .

Now, our expression is

From this, we can cancel out the 2's and an x from both sides.

Thus our answer becomes:

.

In 2010, in our equation because we have had no years past 2010. Plugging that in to the model equation and solving:

, since anything raised to the power of zero becomes . So the population of fish in 2010 is fish.

In 2015, because 5 years have passed since 2010. Plugging that into our equation and solving gives us

So the population of fish in 2015 is fish. This is an example of exponential decay since the function is decreasing.

Since , the equation simplifies to .

Since the bases are equal, we can then set the exponents equal to each other.

Solving for x in this simple equation gives the correct answer.

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x:

The key to this is that . From here, the equation can be factored as if it were .

and

and

and

Now take the natural log (ln) of the two equations.

and

and