# Video: Solving Logarithmic Equations over the Set of Real Numbers

Solve log (logβ (log_(π₯) 36)) = 0, where π₯ β β.

02:53

### Video Transcript

Solve the equation log of log base two of log base π₯ of 36 equals zero, where π₯ is an element of the set of real numbers.

Weβve been given a logarithmic equation, and weβre looking to find the value of π₯. Notice that there are three logarithms involved. So, weβre going to need to be really careful. But what do we mean by a logarithm? Well, letβs take the general equation log base π of π is equal to π. The logarithm of π is the inverse to the exponent of π. So, letβs raise both sides of this general equation as a power of π such that π to the power of log base π of π equals π to the power of π. Since log base π is the inverse to the exponent of π, π to the power of log base π of π is simply π. We, therefore, say that log base π of π equals π is equivalent to saying π is equal to π to the power of π.

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 π₯ here and log base two here. But what is the base of this logarithm? Well, for now, letβs call it π. So, log base π of some number is equal to zero. We can either raise both sides as a power of π to perform the inverse operation two log base π or simply use our definition. And when we do, weβre simply left with, on the left-hand side, log base two of log base π₯ of 36. This is the equivalent to π in our definition, and thatβs equal to π to the power of zero. Well, thatβs π to the power of π 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 π₯ of 36 equals one. Weβre now going to deal with this log base two. And weβ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 π₯ 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 π₯ of 36 equals two.

We have one more logarithm to deal with. Itβs logarithm base π₯. Now, this time, weβre going to raise both sides as a power of π₯. Or again, we refer back to our definition. Either way, weβre simply left with 36 on the left-hand side. And on the right, we have π₯ squared. We now have a very simple equation in terms of π₯. Weβ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 π not only must be positive, but it cannot be equal to one. And so, weβre interested in the positive square root of 36 only, which is six. And so, weβve solved our logarithmic equation, and we get π₯ is equal to six.