# Question Video: Solving Logarithmic Equations over the Set of Real Numbers Mathematics • 10th Grade

Solve logโ (logโ (7๐ฅ + 194)) = 1, where ๐ฅ โ โ.

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### Video Transcript

Solve the equation log base five of log base three of seven ๐ฅ plus 194 equals one, where ๐ฅ is an element of the set of real numbers.

Here, weโve been given a logarithmic equation. Now, weโre going to need to be a little bit careful because we have two different bases in this equation. Weโve got log base five and log base three. And so letโs recall what we actually mean by log base ๐ of ๐ equals ๐.

Logarithm is the inverse function to exponentiation. Itโs the power to which a number must be raised in order to get some other number. Now, letโs raise both sides of our general form as a power of ๐ such that ๐ to the power of log base ๐ of ๐ equals ๐ to the power of ๐. Now, since a logarithm is the inverse function to exponentiation, ๐ to the power of log base ๐ of ๐ is simply ๐. And so log base ๐ of ๐ equals ๐ is equivalent to saying that ๐ is equal to ๐ to the power of ๐.

So letโs go back to the equation in our question. Weโve got log base five of some other algebraic expression. So letโs raise both sides of our equation as a power of five. When we do so on the left-hand side, weโre simply left with log base three of seven ๐ฅ plus 194. In our general form, this is the equivalent to ๐. Raising the right-hand side as a power of five, and when we get five to the power of one. Now, in our general form, thatโs ๐ to the power of ๐. Now, of course, five to the power of one is just five. So our equation becomes log base three of seven ๐ฅ plus 194 equals five.

Now, we have a logarithmic equation purely base three. So to solve, weโre going to raise both sides as a power of three. This time, when we do this, weโre left with seven ๐ฅ plus 194. And again, this corresponds to the letter ๐ in our general form. On the right-hand side, we get three to the fifth power. Now, three to the fifth power is 243. So weโre actually left with quite a simple equation in ๐ฅ. Itโs seven ๐ฅ plus 194 equals 243.

Letโs solve this equation by subtracting 194 from both sides. That gives us seven ๐ฅ equals 49. Finally, we divide through by seven, and we get ๐ฅ is equal to seven. And so, weโve solved the equation log base five of log base three of seven ๐ฅ plus 194 equals one. ๐ฅ is equal to seven. And, of course, itโs useful to recall that we can check our answer as we would check the solution to any other equation by substituting ๐ฅ equals seven back into the original expression. And we should indeed get one.