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Video: Solving Logarithmic Equations

Tim Burnham

If log₉ 𝑥 + log₉ 9 = 2, what is the value of 𝑥?

02:32

Video Transcript

If log base nine of 𝑥 plus log base nine of nine equals two, what is the value of 𝑥?

Okay, before we launch into this question, let’s just take a moment to recall the format of terms involving logarithms. So remember, log base 𝑎 of 𝑏 equals 𝑥 means 𝑎 to the 𝑥 exponent equals 𝑏. Well this 𝑏 here, log base 𝑎 of 𝑏, means which exponent of 𝑎 gives us 𝑏. And in this case, the answer would be 𝑥. 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, that’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 𝑥 plus one is equal to two. And if I subtract one from each side of that equation, I’ve got log base nine of 𝑥 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 𝑥. And nine to the power of one is just nine. So there we have it; our answer is: 𝑥 equals nine.

But before we go, let’s just look at a slightly different way of tackling this question. Looking back at our original question, in both cases here, we’ve got the same base of nine. Now we can use the addition rule of logs to rewrite that expression. Now remember, log base 𝑎 of 𝑥 plus log base 𝑎 of 𝑦 is equal to log base 𝑎 of 𝑥 times 𝑦. So log base nine of 𝑥 plus log base nine of nine could be rewritten as log base nine of 𝑥 times nine or nine 𝑥. And that is equal to two. And this, in turn, is telling us that nine with an exponent of two is equal to nine 𝑥. So nine 𝑥 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 𝑥 equals nine as my answer.

So it’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.