SOLUTION: I need help with this question: A simple technique that biologists use to estimate the age of an African elephant is to measure the length of the elephant's footprint and then c

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Question 718219: I need help with this question:
A simple technique that biologists use to estimate the age of an African elephant is to measure the length of the elephant's footprint and then calculate its age using the equation L = 45 - 257e^-.09a , where L is the length of the elephant's footprint (in centimeters) and a is the age (in years) of the elephant.
Write an equation and solve to find the ages of the elephants whose footprints are 24 cm long, 28 cm long, 32 cm long, and 36 cm long. Then solve the equation L = 45 = 257e^-.09a for a, and use this equation to find the ages of the elephants whose footprints are given above.

For the footprints, I did the equations:
24 cm ; 24 = 45 -257e^-.09a
28 cm; 28 = 45 -257e^-.09a
32 cm; 32 = 45 -257e^-.09a
and 36 cm; 36 = 45 -257e^-.09a
I am not sure how to solve for a using log/natural log/ e etc. Can you please explain this and the process/ how you got the answer? Thanks!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one and leave the rest for you to do. The steps for each problem are exactly the same. The only difference is the number that starts on the left side of the equation,

24 cm ; 24+=+45+-257e%5E%28-.09a%29
First we isolate the base and its exponent. Subtracting 45:
-21+=+-257e%5E%28-.09a%29
Dividing by -257:
-21%2F%28-257%29+=+e%5E%28-.09a%29
Converting the fraction to a decimal:
0.081712062256809338521400778210117+=+e%5E%28-.09a%29
(Feel free to round this decimal off as you choose.) Next we use base e, ln, logarithms (because the base of the exponent is e and because our calculators "know" ln's).
ln%280.081712062256809338521400778210117%29+=+ln%28e%5E%28-.09a%29%29
Next we use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, which allows us to move the exponent of the argument of a log out in front of the log. (It is this property that is the very reason we use logarithms when the unknown is in an exponent. This property let's us move the exponent (with the variable) out in front where we can "get at it" with "regular" algebra.) Using ths property we get:
ln%280.081712062256809338521400778210117%29+=+%28-.09a%29%2Aln%28e%29
Since the base of ln is e, the log on the right is just a 1:
ln%280.081712062256809338521400778210117%29+=+-.09a
Last we divide by -0.09:
ln%280.081712062256809338521400778210117%29%2F%28-.09%29+=a
Using our calculator to find the ln on the left:
%28-2.5045536471717968018511963343956%29%2F%28-.09%29+=a
Dividing:
27.828373857464408909457737048839+=+a
So the elephant is approximately 27.8 years old.