Question 718219
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,<br>
24 cm ; {{{24 = 45 -257e^(-.09a)}}}
First we isolate the base and its exponent. Subtracting 45:
{{{-21 = -257e^(-.09a)}}}
Dividing by -257:
{{{-21/(-257) = e^(-.09a)}}}
Converting the fraction to a decimal:
{{{0.081712062256809338521400778210117 = e^(-.09a)}}}
(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(0.081712062256809338521400778210117) = ln(e^(-.09a))}}}
Next we use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, 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(0.081712062256809338521400778210117) = (-.09a)*ln(e)}}}
Since the base of ln is e, the log on the right is just a 1:
{{{ln(0.081712062256809338521400778210117) = -.09a}}}
Last we divide by -0.09:
{{{ln(0.081712062256809338521400778210117)/(-.09) =a}}}
Using our calculator to find the ln on the left:
{{{(-2.5045536471717968018511963343956)/(-.09) =a}}}
Dividing:
{{{27.828373857464408909457737048839 = a}}}
So the elephant is approximately 27.8 years old.