### Video Transcript

Find the solution set of ๐ฅ to the power of log base ๐ฅ of ๐ฅ to the sixth power equals log of 10 to the power of 64.

Here we have a rather nasty-looking logarithmic equation. Letโs begin by seeing if thereโs any way we can simplify it somewhat. Well, first, we recall one of our laws of logarithms. And this says that log of ๐ to the power of ๐ is the same as ๐ to the power of log ๐. And it doesnโt matter what the base of this log is. This means we can rewrite the right-hand side of our equation as 64 log of 10.

Now, if no base is included, we can assume that the base of our logarithm is 10. So this is actually 64 log base 10 of 10. But we know that log base ๐ of ๐ is simply one. So 64 log base 10 of 10 or 64 log of 10 is 64 times one, which is just 64. And so, weโve simplified the right-hand side of our equation. We get ๐ฅ to the power of log base ๐ฅ of ๐ฅ to the sixth power equals 64. But can we simplify the left-hand side?

Well, weโre actually going to use the same rules. This time, weโre simply going to consider the exponent. Itโs log base ๐ฅ of ๐ฅ to the sixth power. Using our first rule, we see that log base ๐ฅ of ๐ฅ to the sixth power is the same as six log base ๐ฅ of ๐ฅ. Then, using our second rule, we see that log base ๐ฅ of ๐ฅ is one. So six log base ๐ฅ of ๐ฅ is six times one, which is simply six. And so, our equation becomes ๐ฅ to the sixth power equals 64.

Now, to solve for ๐ฅ, we could take the sixth root of both sides, remembering, of course, that we take both the positive and negative sixth root of 64. But actually, we know that two to the sixth power is equal to 64. So ๐ฅ is equal to two. Weโre not actually going to include the solution ๐ฅ is equal to negative two. And thatโs because the base of a logarithm cannot be negative.

And if we go back to our question, ๐ฅ is indeed the base of one of our logs. And so, the solution to this equation is simply ๐ฅ is equal to two. Using set notation, we say that the solution set of ๐ฅ to the power of log base ๐ฅ of ๐ฅ to the sixth power equals log of 10 to the power of 64 is two.