# Video: Solving Logarithmic Equations

If logβ π₯ + logβ 9 = 2, what is the value of π₯?

02:31

### Video Transcript

If log base nine of π₯ plus log base nine of nine equals two, what is the value of π₯?

Okay, before we launch into this question, letβs just take a moment to recall the format of terms involving logarithms. So remember, log base π of π equals π₯ means π to the π₯ exponent equals π. Well this π here, log base π of π, means which exponent of π gives us π. And in this case, the answer would be π₯. So when answering this question, this term here, log base nine of nine, means what exponent do I have to raise the base of nine to in order to get nine. Well, thatβs one because nine with an exponent of one, or nine to the power of one, is equal to nine.

So we can rewrite our equation as log base nine of π₯ plus one is equal to two. And if I subtract one from each side of that equation, Iβve got log base nine of π₯ is equal to one. And this is telling us that nine with an exponent of one, or nine to the power of one, is equal to π₯. And nine to the power of one is just nine. So there we have it; our answer is: π₯ equals nine.

But before we go, letβs just look at a slightly different way of tackling this question. Looking back at our original question, in both cases here, weβve got the same base of nine. Now we can use the addition rule of logs to rewrite that expression. Now remember, log base π of π₯ plus log base π of π¦ is equal to log base π of π₯ times π¦. So log base nine of π₯ plus log base nine of nine could be rewritten as log base nine of π₯ times nine or nine π₯. And that is equal to two. And this, in turn, is telling us that nine with an exponent of two is equal to nine π₯. So nine π₯ is equal to nine squared or nine to the power of two. And nine squared is 81. Then dividing both sides of my equation by nine, again gives me π₯ equals nine as my answer.

So itβs always good that you have another method to approach the question so you can check your answer and make sure you get the same result.