Video Transcript
What is the value of log to the base root two of 32 root two?
So with this expression, the first thing we want to do is do a bit of simplification. And to do that, what we’re gonna do is convert everything into index form. Well, the first thing we can do is use one of our exponent rules. And that rule tells us that if we have root 𝑎, this is equal to 𝑎 raised to the power of a half. So therefore, we’ve got log to the base two raised to the power of a half.
Then next, we can look at 32. Well, as we’ve got the other things that are gonna be two to the power of something, so therefore we need to change 32 to two to the power of something. What we can see that 32 is two multiplied by two multiplied by two multiplied by two multiplied by two. So therefore, we can write it as two raised to the power of five. And then, it’s gonna be multiplied by two to the power of a half. And that’s because if we have a look at root two, we use the same rule that we used before. And we can see that it’s two to the power of a half. Okay, great. So now what’s the next step?
So now, what we can do is we can simplify two to the power of five multiplied by two to the power of a half. And that’s gonna be two to the power of 11 over two. And that’s because we can use one of our exponent laws. And that is if we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is the same as 𝑥 to the power of 𝑎 plus 𝑏. So if we add five to a half, we get five and a half, which is the same as 11 over two or eleven-halves.
So now, in order to solve and find out what the value of our expression is, we’re gonna use one of the basic rules for logarithms. Notice if we have log to the base 𝑎 of 𝑏 is equal to 𝑥 and we convert into exponent form, we can say that 𝑎 to the power of 𝑥 is equal to 𝑏. So therefore, we can say that if we set our is equal to 𝑥, we’ll have log to the base two to the power of a half of two to power of 11 over two is equal to 𝑥. So then, in exponent form, we get two to the power of a half all to the power of 𝑥 is equal to two to the power of 11 over two.
So now, we’re gonna use another exponent rule. And that is if we have 𝑥 to the power of 𝑎 to the power of 𝑏, this is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. So therefore, we get two to the power of 𝑥 over two is equal to two to the power of 11 over two. So now, with two of the same basis, we can equate the exponents. So when we do that, we get 𝑥 over two is equal to 11 over two. So then, if we multiplied both sides by two, we’re gonna get 𝑥 is equal to 11. So therefore, we’ve solved the problem. And we can say that the value of log to the base root two of 32 root two is 11.
Okay, so we did that with one method. I’m just gonna show you another method we could’ve used cause we could’ve have approached in a slightly different way. So with this other method, what we do is we have our expression. And then, we set it equal to 𝑥. Then, we apply our general rule for logarithms. And we get root two to the power of 𝑥 is equal to 32 root two. Then next, what we do is apply the same rule that we started with with the other method, which means we’d have two to the power of a half to the power of 𝑥 equals two to the power of 11 over two. And then, from here, we’d follow the last stages of the previous method. So we’d have two to the power of 𝑥 over two is equal to two to the power of 11 over two. Well, it should give us 𝑥 over two is equal to 11 over two. Then, we finally get 𝑥 is equal to 11, which is the same that we got with the first method.
So we know the value of our expression is 11.
