WEBVTT
00:00:01.240 --> 00:00:07.610
If log base nine of π₯ plus log base nine of nine equals two, what is the value of π₯?
00:00:09.180 --> 00:00:14.680
Okay, before we launch into this question, letβs just take a moment to recall the format of terms involving logarithms.
00:00:15.810 --> 00:00:22.640
So remember, log base π of π equals π₯ means π to the π₯ exponent equals π.
00:00:24.020 --> 00:00:29.480
Well this π here, log base π of π, means which exponent of π gives us π.
00:00:30.670 --> 00:00:32.840
And in this case, the answer would be π₯.
00:00:34.320 --> 00:00:44.360
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.
00:00:46.020 --> 00:00:51.930
Well, thatβs one because nine with an exponent of one, or nine to the power of one, is equal to nine.
00:00:52.960 --> 00:00:58.160
So we can rewrite our equation as log base nine of π₯ plus one is equal to two.
00:00:59.200 --> 00:01:04.950
And if I subtract one from each side of that equation, Iβve got log base nine of π₯ is equal to one.
00:01:06.170 --> 00:01:12.810
And this is telling us that nine with an exponent of one, or nine to the power of one, is equal to π₯.
00:01:14.510 --> 00:01:17.560
And nine to the power of one is just nine.
00:01:18.790 --> 00:01:22.190
So there we have it; our answer is: π₯ equals nine.
00:01:23.820 --> 00:01:27.200
But before we go, letβs just look at a slightly different way of tackling this question.
00:01:28.640 --> 00:01:33.900
Looking back at our original question, in both cases here, weβve got the same base of nine.
00:01:35.250 --> 00:01:38.670
Now we can use the addition rule of logs to rewrite that expression.
00:01:39.380 --> 00:01:46.040
Now remember, log base π of π₯ plus log base π of π¦ is equal to log base π of π₯ times π¦.
00:01:47.280 --> 00:01:55.620
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₯.
00:01:56.860 --> 00:01:58.670
And that is equal to two.
00:01:59.850 --> 00:02:05.370
And this, in turn, is telling us that nine with an exponent of two is equal to nine π₯.
00:02:06.890 --> 00:02:11.770
So nine π₯ is equal to nine squared or nine to the power of two.
00:02:12.840 --> 00:02:15.080
And nine squared is 81.
00:02:16.580 --> 00:02:21.720
Then dividing both sides of my equation by nine, again gives me π₯ equals nine as my answer.
00:02:22.900 --> 00:02:29.270
So itβs 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.