WEBVTT
00:00:01.330 --> 00:00:19.470
Given that log three to the base seven is approximately equal to 0.5646, find, without using a calculator, the value of log 147 to the base seven correct to four decimal places.
00:00:21.050 --> 00:00:31.120
Before starting this question, we need to recall one of our laws of logarithms: log π₯ plus log π¦ is equal to log π₯π¦.
00:00:32.680 --> 00:00:37.960
147 is equal to three multiplied by 49.
00:00:38.980 --> 00:00:49.290
This means that log of 147 to the base seven can be rewritten as log of three multiplied by 49 to the base seven.
00:00:50.430 --> 00:01:00.390
Using the law that we have quoted, this in turn can be rewritten as log three to the base seven plus log 49 to the base seven.
00:01:01.530 --> 00:01:10.910
Weβre told in the question that log three to the base seven is approximately equal to 0.5646.
00:01:11.960 --> 00:01:14.860
49 is equal to seven squared.
00:01:15.260 --> 00:01:23.140
Therefore, log 49 to the base seven can be rewritten as log of seven squared to the base seven.
00:01:24.470 --> 00:01:34.010
At this point, we need to consider another one of the laws of logarithms: log π₯ to the base π is equal to π log π₯.
00:01:35.160 --> 00:01:40.650
We can, therefore, rewrite log of seven squared as two log seven.
00:01:42.110 --> 00:01:45.470
Log of π to the base π is equal to one.
00:01:46.540 --> 00:01:51.610
This means that log of seven to the base seven will also be equal to one.
00:01:52.760 --> 00:02:01.120
Our expression, therefore, simplifies to 0.5646 plus two multiplied by one.
00:02:02.190 --> 00:02:08.850
As two multiplied by one is two, weβre left with 2.5646.
00:02:09.750 --> 00:02:20.290
The value of log 147 to the base seven correct to four decimal places is 2.5646.