If the population continues to decline at this rate, to what fraction of the original population size will the population decline after 10 years?

(Keep at least 4 significant digits.)

##### Hint

If the population is decreasing at a rate of 5% per year, then by what number do we need to multiply the population size each year?

We didn't tell you how many sea lions there were to start out with, so apparently, your answer shouldn't depend on that value. If you like, you can define a variable to represent the initial population size (for example $p_0$); then, you can multiply that initial population size by a certain number once for each year that passes. However, since we are asking for what fraction of the original population size is left (for example, what fraction of the original $p_0$), you need to divide by that original population size to get your answer. In the end, the variable you chose for initial population size should drop out.

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To what fraction of the original population size will the population decline after $n$ years?

(Online, enter exponentiation using `^`, so enter $a^b$ as `a^b`.)

##### Hint

Same procedure as for the previous part, except that you have multiply by that number $n$ times.

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To find how long it will take for the population to decline by one-half, follow these steps. Set the expression from part (b) equal to one-half.

$ = \frac{1}{2}$

Take the logarithm of both sides of that equation.

=

(Online, you can use either `ln` or `log` for logarithm; both are interpreted a logarithm base $e$. In this case, it doesn't matter what base logarithm you use. Also, don't simplify your answers yet, but leave them as the logarithm of your previous answers.)

Using the log of power rule, bring down the exponent from the left hand side in front of the logarithm.

=

Solve for the value of $n$. The result is a ratio of logarithms.

$n=$

$\approx$

(In the first blank, write the ratio of logarithms. In the second blank, give a decimal approximation with at least 4 significant digits.)

To repeat, how long will it take for the population to decline by one-half?

(The second blank is for a unit.)

You can use a similar procedure to find out how long it will take for the population to decline to one-tenth its original size. Set the expression from part (b) equal to one-tenth.

$ = \frac{1}{10}$

Take the logarithm of both sides of that equation.

=

(Online, you can use either `ln` or `log` for logarithm; both are interpreted a logarithm base $e$. In this case, it doesn't matter what base logarithm you use.)

Using the log of power rule, bring down the exponent from the left hand side in front of the logarithm.

=

Solve for the value of $n$. The result is a ratio of logarithms.

$n=$

$\approx$

(In the first blank, write the ratio of logarithms. In the second blank, give a decimal approximation with at least 4 significant digits.)

To repeat, how long will it take for the population to decline to one-tenth of its original size?

(The second blank is for a unit.)