Solve the equation log of log base two of log base 𝑥 of 36 equals zero, where 𝑥 is an element of the set of real numbers.
We’ve been given a logarithmic equation, and we’re looking to find the value of 𝑥. Notice that there are three logarithms involved. So, we’re going to need to be really careful. But what do we mean by a logarithm? Well, let’s take the general equation log base 𝑏 of 𝑎 is equal to 𝑐. The logarithm of 𝑏 is the inverse to the exponent of 𝑏. So, let’s raise both sides of this general equation as a power of 𝑏 such that 𝑏 to the power of log base 𝑏 of 𝑎 equals 𝑏 to the power of 𝑐. Since log base 𝑏 is the inverse to the exponent of 𝑏, 𝑏 to the power of log base 𝑏 of 𝑎 is simply 𝑎. We, therefore, say that log base 𝑏 of 𝑎 equals 𝑐 is equivalent to saying 𝑎 is equal to 𝑏 to the power of 𝑐.
We can either use this definition or use the fact that the log base is the inverse operation to the exponential. We do have a little bit of a problem though. We got log base 𝑥 here and log base two here. But what is the base of this logarithm? Well, for now, let’s call it 𝑏. So, log base 𝑏 of some number is equal to zero. We can either raise both sides as a power of 𝑏 to perform the inverse operation two log base 𝑏 or simply use our definition. And when we do, we’re simply left with, on the left-hand side, log base two of log base 𝑥 of 36. This is the equivalent to 𝑎 in our definition, and that’s equal to 𝑏 to the power of zero. Well, that’s 𝑏 to the power of 𝑐 in our definition.
We, of course, know that anything to the power of zero is one. So, our equation becomes log base two of log base 𝑥 of 36 equals one. We’re now going to deal with this log base two. And we’re going to deal with this by either raising both sides as a power of two or going back to our definition. When we do, the left-hand side simply becomes log base 𝑥 of 36. Remember, this is because the logarithm with a base of two is the inverse operation to the exponent of two. On the right-hand side, we get two to the power of one, which is just two. So, our new equation is log base 𝑥 of 36 equals two.
We have one more logarithm to deal with. It’s logarithm base 𝑥. Now, this time, we’re going to raise both sides as a power of 𝑥. Or again, we refer back to our definition. Either way, we’re simply left with 36 on the left-hand side. And on the right, we have 𝑥 squared. We now have a very simple equation in terms of 𝑥. We’re going to find the square root of both sides of our equation.
Now, usually we would find both the positive and negative square root of 36. But 𝑏 not only must be positive, but it cannot be equal to one. And so, we’re interested in the positive square root of 36 only, which is six. And so, we’ve solved our logarithmic equation, and we get 𝑥 is equal to six.