Video: Rearranging Formulas Involving Logarithms

In this video, we will learn how to rearrange formulas that contain logarithms and exponents.

13:15

Video Transcript

In this video, our topic is rearranging formulas involving logarithms. By the end of the lesson, we’ll understand just how it is that the left-hand side of this equation we see on screen, as long and as complicated as it looks, really does equal simply one.

As we get started working with logarithms, let’s consider this expression here, π‘Ž is equal to two raised to the 𝑏. Given this expression, we know that if we take the log base two of this number π‘Ž, then the result we’ll get is 𝑏. But then notice this. Since π‘Ž is equal to two raised to the 𝑏, we could substitute that in for π‘Ž in our logarithm equation. And that would give us this expression, the log base two of two raised to the 𝑏 equals 𝑏. Here, we have an exponential function two raised to the 𝑏, and this is the argument of a log base two function. And we see that with the base of our logarithm and the base of our exponent being the same, the result of this whole calculation on the left is simply the exponent all by itself.

We could say then that our logarithm and our exponent are canceling one another out, that they’re inverses. And it turns out that this is true in general, regardless of the base that we’re working with. Mathematically, we can express that fact this way. We can say that a logarithm function with a given base π‘₯, when this is applied to an exponential function with a base π‘₯ and an exponent 𝑦, will invert that function. And the result will be the exponent 𝑦. Notice that this example here, with a base of two for the log and the exponent, is a special case of this general relation. It’s important to note that, for this relation to hold, the base of our exponential function and the base of our logarithmic function must be the same.

We can use this fact to solve for certain variables in mathematical equations. For example, say that we were given this equation here and we wanted to make the variable 𝑦 the subject. Given this task, we might first think to divide both sides of the equation by π‘Ž, whatever that value is. That would give us this result. But now, how do we bring 𝑦 downstairs, so to speak, out of the exponent of this function? We can recall that because logarithms and exponents are inverses, then if we take the logarithm, base 𝑏, of this exponent, we’ll wind up with just that exponent 𝑦. Notice that in order to apply this logarithm here and maintain our mathematical equality, we have to apply it to the other side of our equation too.

So then, by this rule here, the log base 𝑏 of 𝑏 to the y simplifies to 𝑦. And now we have this variable as the subject of our equation. We start to see then that logarithms as well as exponents, since they’re inverses, can be useful tools for helping us rearrange formulas. To see another example of how this might work, let’s consider this equation here, π‘Ÿ is equal to 𝑒 raised to the 𝑠 plus 𝑑. We can recall that this particular exponential function, this letter 𝑒 raised to a value, is called the exponential function, where 𝑒 has a numerical value of approximately 2.71. So then, given this equation, how would we make 𝑠, the exponent of this exponential function, the subject?

The first thing we might think to do is to subtract 𝑑 from both sides so that it cancels on the right side. But then, what can we do next to bring 𝑠 downstairs and make it the subject? Recalling once again that logarithms and exponents are inverses, we can realize that the exponential function, which is an exponent with a particular value for its base, has as its inverse a particular logarithmic function called the natural logarithm.

The fact that these two functions, the exponential function and the natural logarithm, are inverses means that according to our rule here, if we were to replace π‘₯ to the 𝑦 with 𝑒 to the π‘₯ and the logarithm base π‘₯ with the natural logarithm, then the left-hand side of our expression would look like this, the natural logarithm of 𝑒 to the π‘₯. And this is equal to the exponent π‘₯.

When we say then that the exponential function and the natural logarithm are inverses, this is part of what that means mathematically. We say part of what it means because really we could write this in the opposite order. Here we have the natural logarithm of an exponential function. But we could equivalently take an exponential function and have as its exponent the natural logarithm of π‘₯. And once again because these two functions are inverses of one another, this is equal to π‘₯.

All this to say when we come back over here trying to make 𝑠 our subject, if we apply the natural logarithm to both sides of the expression, that looks like this. And then, since the natural logarithm of 𝑒 raised to the π‘₯ is equal simply to π‘₯, we can then say that the natural logarithm of 𝑒 to the 𝑠 simplifies to 𝑠 itself. As we work with these expressions involving exponents and logarithms, it’s worthwhile to recall some rules for working with those functions.

Starting with exponents, we can remember that if we have some number raised to the power π‘₯ multiplied by that same number raised to a different power, 𝑦, then that product is equal to that base, that particular number we’ve called π‘Ž, to the π‘₯ plus 𝑦 power. And then, if we consider that the opposite of multiplication is division and the opposite of addition is subtraction, we come to understand the second rule for working with exponents. And then lastly, if we have some base π‘Ž raised to the power π‘₯ and that whole expression is raised to the power 𝑦, then that’s equal to that base π‘Ž to the π‘₯ times 𝑦 power.

Along with these rules for exponents, we can recall some corresponding ones for logarithms. First off, that a log with a given base of a product of two numbers, π‘₯ times 𝑦, is equal to the sum of the log with that same base of those two separate values, π‘₯ and 𝑦. Similarly, the log base π‘Ž of a fraction, π‘₯ divided by 𝑦, is equal to the log base π‘Ž of the numerator minus the log base π‘Ž of the denominator. And lastly, if we have a logarithm of an exponent and notice that in this general case the base of our log is not the same as the base of our exponent, then this is equal to that exponent times the log base π‘Ž of the base of the exponential function π‘₯.

We can see in this last rule here a general expression that we found a specific expression for over here. That is, when the base of our logarithm and the base of our exponential function are equal, then this expression on the left simplifies to the exponent itself. And over here, in our very general rule, if we were taking the log base π‘Ž of π‘₯, where π‘₯ was equal to π‘Ž, then all of this would be equal to one. And we would find a result that agrees with this expression here. Now, our main focus here is not the rules themselves, but rather to let them help us rearrange formulas involving logarithms and exponents.

Knowing all this, let’s clear a bit of space at the top of our screen and look once more at that expression we found on the opening screen of this video. That expression looked like this. And now we can analyze the left-hand side to see if it really does equal what we’ve said it equals. We’ll start at the most interior mathematical function. That’s this exponential function here, 𝑒 raised to the natural log of one. We’ve seen that 𝑒 raised to the natural log of some number π‘₯ is equal to π‘₯ because the exponential function and the natural logarithm are inverses. So that means that all of this is equal to one.

Next, we can look at this exponential function, which is 𝑏 raised to the log base 𝑏 of one. To see what this equals, let’s look down here for a moment. We said that taking the natural logarithm of an exponential function inverts those functions in the same way that taking the exponent of a natural logarithm does. And we recall further that the exponential function 𝑒 and the natural logarithm 𝑙𝑛 are simply an exponent and a log with a particular base value. That tells us that this expression here is a specific application of this more general rule.

In this rule, we see that we’re first applying a logarithmic function to an exponential one. But since logarithms and exponents are inverses, we could equally well do it the opposite way, use an exponential function first and take as its argument a logarithm. Using these variables, that would look like this: π‘₯, the base of our exponential function, raised to the log base π‘₯ of 𝑦. And this too is equal to 𝑦.

The important equality here is that the base of our logarithmic function, π‘₯, is the same as the base of our exponential one. When that’s true, then these equalities all hold. As we look at this part of our expression, we can notice that it applies to 𝑏 raised to the log base 𝑏 of one. Here, we’ve got an exponential expression with a base 𝑏 equal to the base of our logarithm. And so what this all simplifies down to is the value in the parentheses of that logarithm, that is, one.

And now we’ll consider the natural logarithm of 𝑒 raised to the first power. Looking down at our rule here, the natural logarithm of 𝑒 raised to any number is equal to that number. So therefore, this is all one. And now on the left-hand side, we have the logarithm base π‘Ž of π‘Ž to the one. If we apply this rule here, that the log base π‘₯ of π‘₯ to the 𝑦 is equal to 𝑦, then we see that all of this expression will simplify down to the exponent here. And so all of that left-hand side really does equal one. Let’s get a bit more practice now with these ideas through an example exercise.

The quantities π‘Ž, 𝑏, and 𝑐 are related to each other by the formula π‘Ž 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 π‘Ž minus 𝑏. (B) 𝑐 is equal to 𝑒 raised to the 𝑏 divided by π‘Ž. (C) 𝑐 is equal to the natural logarithm of 𝑏 divided by π‘Ž. (D) 𝑐 is equal to the natural logarithm of π‘Ž divided by 𝑏. (E) 𝑐 is equal to 𝑒 raised to the π‘Ž divided by 𝑏 power.

So given these quantities π‘Ž, 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 π‘₯, if we take the natural logarithm of that exponential function, then the result is the exponent π‘₯. 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 π‘Ž divided by 𝑏. Scanning through our different answer options, we see that option (D) agrees with this result.

So if we have three quantities π‘Ž, 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 π‘Ž 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 π‘Ž of π‘Ž raised to the π‘₯ power is equal to π‘₯ and, equivalently, so is π‘Ž raised to the log base π‘Ž of π‘₯. We saw further that a specific case of this involves the exponential function, 𝑒 to the π‘₯, and the natural logarithm, 𝑙𝑛 of π‘₯.

And lastly, we saw that when they’re applied to mathematical equations, logarithms and exponents operate on all terms on both sides of an equation. In this way, applying a logarithmic or exponential function is like multiplying by a constant value all through an equation. Just as that multiple must be applied to every term, so must the logarithm or exponent applied. This is a summary of rearranging formulas involving logarithms.

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