### Video Transcript

Solve the equation log base three of log base four of ๐ฅ squared plus 39 equals one, where ๐ฅ is an element of the set of all real numbers.

Weโve been given a logarithmic equation for which we need to solve for ๐ฅ. Weโll need to be really careful here. We have two different bases. Weโve got base three and base four for our logarithms. So what is a logarithm? Well, itโs simply another mathematical operation. This time, it determines how many times a certain number โ โ thatโs the base โ โ is multiplied by itself to reach another number.

Letโs take the general equation log base ๐ of ๐ equals ๐. We raise both sides as a power of ๐. And this is really useful because logarithms are the inverse operation to exponentiation. So ๐ to the power of log base ๐ of ๐ is simply ๐. And on the right-hand side, we get ๐ to the power of ๐. So these two equations are equivalent. Log base ๐ of ๐ equals ๐ is the same as saying ๐ is equal to ๐ to the power of ๐. So letโs go back to our equation. Weโve got log base three of some expression. Weโre going to raise both sides of our equation then as a power of three.

When we do, the left-hand side becomes log base four of ๐ฅ squared plus 39. Thatโs the equivalent of ๐ in our general form. The right-hand side becomes three to the power of one. Thatโs the equivalent of ๐ to the power of ๐. But, of course, three to the power of one is simply three. So our equation is log base four of ๐ฅ squared plus 39 equals three. Notice we have another logarithmic equation. This time, itโs base four, so weโre now going to raise both sides as a power of four. Four to the power of log base four of ๐ฅ squared plus 39 is just ๐ฅ squared plus 39. And our right-hand side becomes four cubed. Of course, we know four cubed is equal to 64, so we have ๐ฅ squared plus 39 equals 64.

Weโre now going to solve for ๐ฅ. Letโs subtract 39 from both sides of our equation such that ๐ฅ squared is equal to 25. And then, weโre going to take the square root of both sides, remembering, of course, we need to take both the positive and negative square root of 25. And when we do, we get ๐ฅ is equal to plus or minus five. And these are real numbers. So there are two solutions to our equation where ๐ฅ is an element of the set of real numbers. Itโs five or negative five.

Now, remember, we could also check this solution by substituting ๐ฅ equals five and ๐ฅ equals negative five into our original equation. In both cases, we do indeed get one, so we know our solutions are correct.