Video Transcript
The quantities 𝑎, 𝑏, and 𝑐 are
related to each other by the formula 𝑎 is equal to 𝑏 times 𝑒 raised to the
𝑐. Which of the following shows a
rearrangement of this formula with 𝑐 as the subject? (A) 𝑐 is equal to the natural
logarithm of 𝑎 minus 𝑏. (B) 𝑐 is equal to 𝑒 raised to the
𝑏 divided by 𝑎. (C) 𝑐 is equal to the natural
logarithm of 𝑏 divided by 𝑎. (D) 𝑐 is equal to the natural
logarithm of 𝑎 divided by 𝑏. (E) 𝑐 is equal to 𝑒 raised to the
𝑎 divided by 𝑏 power.
So given these quantities 𝑎, 𝑏,
and 𝑐, which are related to one another through this formula, what we want to do is
rearrange the formula so that 𝑐 is the subject and see which one of our answer
options that rearrangement agrees with. Our mission, then, is to rearrange
this expression so it reads 𝑐 is equal to some other quantities. The first step we can take to do
that is to divide both sides of the equation by the quantity 𝑏. This cancels that term out on the
right. Our expression then is in this
form. But then, how do we bring 𝑐 down
from this exponent?
We can do this by recalling that
the exponential function, 𝑒 raised to some value, has a corresponding inverse
function, that is, a function that undoes the exponential operation. This inverse function is called the
natural logarithm. The way it works, if we have an
exponential function of some variable 𝑥, if we take the natural logarithm of that
exponential function, then the result is the exponent 𝑥. In our example, the exponent we’re
interested in is 𝑐. So if we’d apply the natural
logarithm to 𝑒 raised to the 𝑐, then this rule tells us we can isolate 𝑐.
An important thing to remember
though about applying the natural logarithm function is that it’s an operation
that’s like multiplication or division. In other words, if we apply the
natural logarithm to one side of our equation, then we need to apply it to the other
side as well. So that’s what we’ll do. Here, we’ve applied that natural
logarithm to both sides, and on the right, we have the natural log of 𝑒 to the
𝑐. As we said, by applying this rule
here, this right-hand side simplifies to the exponent itself. And that leaves us with this
expression here, 𝑐 is equal to the natural logarithm of 𝑎 divided by 𝑏. Scanning through our different
answer options, we see that option (D) agrees with this result.
So if we have three quantities 𝑎,
𝑏, and 𝑐 related to one another according to this equation here, then a
rearrangement of this formula with 𝑐 as the subject is 𝑐 is equal to the natural
logarithm of 𝑎 divided by 𝑏.
Let’s summarize now what we’ve
learned about rearranging formulas involving logarithms. In this lesson, we learned that
logarithms and exponents are inverse functions. Mathematically, this means that the
logarithm base 𝑎 of 𝑎 raised to the 𝑥 power is equal to 𝑥 and, equivalently, so
is 𝑎 raised to the log base 𝑎 of 𝑥. We saw further that a specific case
of this involves the exponential function, 𝑒 to the 𝑥, and the natural logarithm,
𝑙𝑛 of 𝑥.
