Hi. You just need more practice using logarithms and working with exponential growth/decay. (Don't feel bad, if you can't understand what Jomo did in post #6. Some of his results do not make sense.)
This is the equation we need to solve for k:
4e^(7k) = 6
Note that k is currently in the exponent position. We need to get k out of the exponent position (our goal is write k=expression, not e^k=expression). We're going to use a basic property of logarithms, to get k out of the exponent position.
We first isolate the exponential term e^(7k), on the left-hand side. We do that by dividing each side of the equation above by 4:
e^(7k) = 3/2
Now that we have the exponential term by itself, we take the natural logarithm of each side:
ln(e^[7k]) = ln(3/2)
Now we're ready to apply a basic property of logarithms. (You're expected to memorize this property and understand how to use it. One of the reasons you're struggling is because you haven't practiced enough using logarithms and properties to recall when and how to use them.)
Here is the basic property of logarithms that we need right now:
ln(b^n) = n · ln(b)
On the left-hand side, we see the natural logarithm of a power (that power is b^n). This property tells us -- when we take the logarithm of a power -- that we may move the exponent n out front as a factor, while changing the exponent on b to 1. In other words, it allows us to get n out of the exponent position.
In the equation we're trying to solve, we took the natural logarithm of a power:
ln(e^[7k])
We apply the property and write:
7k · ln(e)
The equation we're trying to solve for k now looks like this:
7k · ln(e) = ln(3/2)
You're expected to have also memorized that the expression ln(e) always represents 1 (and understand why). Therefore, we replace the expression ln(e) above with the number 1.
7k = ln(3/2)
Finish solving for k. Then replace symbol k with your result below.
N = 4e^(k·t)
You can now find N, when t is 30, by setting t=30 above and evaluating the right-hand side.
PS: Are you currently enrolled in a math class, Tara Marie?
?