Question Video: Finding Values of a Logarithmic Function Mathematics

Video Transcript

Find the missing table values for ℎ of 𝑥 equals log base two of 𝑥.

And then we have a table with three missing values. So, what do we mean when we talk about a logarithm? Logarithmic functions are inverses to exponential functions. Imagine we have the given expression. We read this as log base 𝑏 of 𝑎 equals 𝑐. 𝑏 is the base, 𝑐 is set to be the exponent in this expression, and 𝑎 is called the argument. And in fact, this is exactly the same way as describing the relationship between 𝑎, 𝑏, and 𝑐 as 𝑏 to the power of 𝑐 equals 𝑎. So with this in mind, let’s take the function log base two of 𝑥, and we’ll take our first value of 𝑥, negative two.

Substituting 𝑥 equals negative two in the function for ℎ of 𝑥 gives us ℎ of negative two is equal to log base two of negative two. But we need to work out the value of ℎ of negative two. So let’s call that 𝑐 sub one. If we equivalently describe this relationship as two to the power of 𝑐 sub one equals negative two, we see we need to find the value of 𝑐 sub one that satisfies this equation. But there is no power of two that will give us a value of negative two. This could in fact only be achieved if the base itself was negative. So, 𝑐 sub one is in fact not defined. ℎ of negative two then, we say, is undefined.

Let’s now move on to 𝑥 equals one. ℎ of one in our table will be the value of log base two of one. Defining ℎ of one to be equal to 𝑐 sub two this time, we can equivalently write this relation as two to the power of 𝑐 sub two equals one. To solve this equation, we ask ourselves, what power of two gives an answer of one? Well, in fact, the only way for this to be true is if 𝑐 sub two equals zero. Any real number not equal to the power of zero raised to the power of zero will always be one. So ℎ of one and the second value in our table is zero.

We’re now going to repeat this for 𝑥 equals two. ℎ of two is log base two of two. Defining ℎ of two as 𝑐 sub three, we see that we can represent this relation as two to the power of 𝑐 sub three equals two. So once again, we ask ourselves, what power of two gives two? Well, the only power of two that will give us two itself will be one. So 𝑐 sub three, and in fact ℎ of two, is equal to one. And so we’ve completed the values in our table; they are undefined, zero, and one, respectively.