Problems and Solutions - Natural Log Functions
January 25, 2006
Here are two problems worked out - hopefully, they will help when you are doing
your homework. As mentioned in class, we will be using the natural logarithm
function as a tool to solve problems that involve exponentials. Remember the
rules for using logarithms.
ln(A·B) = ln(A) + ln(B)
ln(Ap) = p·ln(A)
Problem #1: Find the value of tthat solves the following problem.
200 = 120(1.3)t
First, notice that you are solving for a variable that is an exponent.
This should make you think that logarithm function will be helpful
in solving this problem. So, take the natural log of both sides of the
equation.
ln(200) = ln( 120(1.3)t)
Now apply, the product rule from above.
ln(200) = ln( 120 ) + ln( (1.3)t)
Now apply, the exponent rule from above.
ln(200) = ln( 120 ) + tln( 1.3 )
Now you have an equation that you can easily solve. Move the numbers
to one side of the equation and the variables to the other.
ln(200) −ln(120) = tln( 1.3 )
Now, divide both sides by ln(1.3) to solve for t.
t=ln(200) −ln(120)
ln(1.3) =5.2983 −4.7875
.2623 = 1.947
Things to Note
•The function y= 120(1.3)tis exponential growth and has initial value of
120. So, for this function to equal 200 means that t > 0. So, getting an
answer of t= 1.947 makes sense.
•As mentioned in class, ln(200) −ln(120) is NOT equal to ln(80). Use your
calculator to figure out each of these numbers and you will see that the two
are not equal.
•Use at least 4-decimal in your calculations.
