SOLUTION: Garmin's yearly net sales between 2000 and 2008 can be modeled by y=191,525e^0.3162x thousand dollars (sic), where "x" is the number of years since 2000. A. Write the logarithmi

Question 958412: Garmin's yearly net sales between 2000 and 2008 can be modeled by y=191,525e^0.3162x thousand dollars (sic), where "x" is the number of years since 2000.
A. Write the logarithmic form of this equation.
B. Use the logarithmic form to find when Garmin's yearly net sales will reach $3,600,000,000.
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I attempted to do a few different things here, based on lecture notes and my extremely meager understanding of this level of algebra; first, I decided to give myself a base-line to work from. I know that the standard form of the equation should tell me how many years it will take to get to $3.6 billion. I'm slightly confused at the wording of the question, which states "191,525e^0.3162x thousand dollars" which makes me wonder - are they truncating "millions" of dollars into "thousands" to make the numbers more manageable? This seems to be case, since a Google search reveals that Garmin made more than $130 million in 2000, so an equation saying they made $191 million sounds plausible.
So, assuming that this oddly-worded sentence literally wants me to add "thousand" to the end of my numbers, then I can verify that Garmin made $191 million in 2000 by substituting 0 in for "x" (0 representing 2000):
y=191,525e^0.3162(0)
e^0.3162(0)=1 so the equation works out to y=191,525 meaning that in 2000, Garmin made $191,525 thousand dollars, or $191,525,000. So, equation verified.
So to find out how many months it takes to reach $3.6 billion with the standard equation, plug in $3.6 billion (in "thousands") in for y:
3,600,000=191,525e^0.3162x and solve for "x". This evaluates to x=9.27789619 so I know that Garmin will net $3.6 billion sometime in early 2009.

Here's where things get confusing for me. When I write the standard equation in logarithmic form based on what I remember from lecture, I get something like:
ln(191,525)+0.3162x
which is pretty clearly not correct, since by plugging in numbers and evaluating, I get something that isn't even close to 9 years.
So obviously I'm missing an important step. An online calculator tells me that the correct format of the logarithm is ln(y)=(0.3162x).
However, plugging in the numbers for the variables, it doesn't even seem like the log format that the calculator gives me is correct:
ln(y)=(0.3162x) gives me 1 if I plug in 0 for "x" when it should be giving me 191,525, and plugging in 3,600,000 for "y" gets me 47.74334 which isn't even close to 9 years.

I am COMPLETELY lost at this point. I clearly don't understand the correct way to put this equation into logarithmic form, OR how to use the logarithmic form of the equation to solve a simple problem. I would greatly appreciate any help.
Found 2 solutions by rothauserc, Theo:
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The definition of logarithm is
given x = b^y then y = log base b of x
we are given
y=191,525e^0.3162x
note that y and x are interchanged in this example, therefore
0.3162x = ln (y/191,525)
use logarithm rule for division
0.3162x = ln(y) - ln(191,525)
calculate the natural log (ln) of 191,525
A) 0.3162x = ln(y) - 12.162773627
x = (ln(y) - 12.162773627) / 0.3162
now continue calculations, I broke them out so as to clarify process
0.3162x = ln(3,600,000,000) - 12.162773627
calculate the natural log (ln) of 3,600,000,000
0.3162x = 22.004199682 - 12.162773627
0.3162x = 9.841426055
B) x = 31.12405457 approx 31
x represents number of years since 2000, therefore
The year sales reach $3,600,000,000 is 2031

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Garmin's yearly net sales between 2000 and 2008 can be modeled by y=191,525*e^0.3162x thousand dollars (sic), where "x" is the number of years since 2000.
A. Write the logarithmic form of this equation.
B. Use the logarithmic form to find when Garmin's yearly net sales will reach $3,600,000,000.

assuming the present value is 191,525,000 rather than 191,525, the calculations would be as follows:

the equation is:

y = 191,525,000*e^(.3162*x)

x is the number of years since 2000.

the definition of logs tells you:

y = b^x if and only if log_b(y) = x

however, you can solve this problem by simply using the exponential form of the equation and then taking the natural log of both sides of the equation to solve for x.

this is what you do:

start with the equation the way they gave it to you.

y = 191,525,000*e^(.3162*x)

y is the future value
191,525,000 is the present value
.3162 is the annual rate of increase
x is the number of years.

you know what y is and you want to solve for x.

y = 3,600,000,000.

replace y in your equation with 3,600,000,000 and you get:

3,600,000,000 = 191,525,000*e^(.3162*x)

divide both sides of this equation by 191,525,000 to get:

(3,600,000,000 / 191,525,000) = e^(.3162*x)

take the natural log of both sides of this equation to get:

ln(3,600,000,000 / 191,525,000) = ln(e^(.3162*x))

since ln(e^(.3162*x)) is equal to .3162 * x * ln(e), your equation becomes:

ln(3,600,000,000 / 191,525,000) = .3162 * x * ln(e)

since ln(e) is equal to 1, your equation becomes:

ln(3,600,000,000 / 191,525,000) = .3162 * x

divide both sides of this equation by .3162 and you get:

ln(3,600,000,000 / 191,525,000) / .3162 = x

solve for x to get:

x = 9.277896192

the yearly net sales will be equal to 3,600,000,000 in 9.277896192 years at the stated yearly continuous compounding rate as shown by the equation.

confirm by replacing x in your original equation with 9.277896192.

you get:

y = 191,525*e^{(.3162*9.277896192)}

solve for y to get:

y = 3,600,000,000

the solution is good based on the equation.

3,600,000,000 is equal to 3.6 billion dollars.

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i think you went wrong when you translated ln(e^.3162x)

ln(e^.3162x) is equal to .3162x * ln(3) which is equal to .3162x.

the general conversion forms are:

log(a * b) = log(a) + log(b)

log(a / b) = log(a) - log(b)

log(a^b) = b * log(a)

with natural log, you just replace log with ln.

ln(e^b) = b * ln(e) which is equal to b because ln(e) is equal to 1.

why is ln(e) equal to 1?

because ln(e) = y if and only if e^y = e and this occurs only when y = 1.

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