1
00:00:00,000 --> 00:00:04,272
We are asked to solve log of x plus log of 3 is equal to

2
00:00:04,272 --> 00:00:07,082
2 log of 4 minus log of 2.

3
00:00:07,082 --> 00:00:10,170
So let's just rewrite it. So we have the log of x

4
00:00:10,170 --> 00:00:13,259
plus the log of 3 is equal to

5
00:00:13,259 --> 00:00:19,691
2 times log of 4 minus log of 2.

6
00:00:19,691 --> 00:00:23,522
And this is reminder: whenever you see logarithm

7
00:00:23,522 --> 00:00:26,796
written without a base, the implicit base is 10.

8
00:00:26,796 --> 00:00:31,742
So we could write 10 here, here, here and here.

9
00:00:31,742 --> 00:00:34,598
But for the rest of this example, I'll just skip writing 10

10
00:00:34,598 --> 00:00:36,293
just cause it'll save a little bit time.

11
00:00:36,293 --> 00:00:38,731
But remember it only means log base 10.

12
00:00:38,731 --> 00:00:40,263
So this expression right over here

13
00:00:40,263 --> 00:00:43,166
is the power I have to raise 10 to get x.

14
00:00:43,166 --> 00:00:45,883
Power I have to raise 10 to get 3.

15
00:00:45,883 --> 00:00:49,412
Now that out of the way, let's see what logarithm properties we can use.

16
00:00:49,412 --> 00:00:52,988
So we know -- and these are all the same base --

17
00:00:52,988 --> 00:01:00,488
we know that if we have log base a of b plus log base a of c

18
00:01:00,488 --> 00:01:06,432
that this is the same thing as log base a of bc.

19
00:01:06,432 --> 00:01:08,592
And we also know -- so let me write all of the

20
00:01:08,592 --> 00:01:10,449
logarithm properties that we know, over here.

21
00:01:10,449 --> 00:01:14,536
We also know that if we have a logarithm --

22
00:01:14,536 --> 00:01:22,268
let me write it this way actually b times log base a of c

23
00:01:22,268 --> 00:01:26,610
this is equal to log base a of c to the b-th power.

24
00:01:26,610 --> 00:01:29,559
And we also know -- and this is derived really straight from

25
00:01:29,559 --> 00:01:31,463
both of these, is:

26
00:01:31,463 --> 00:01:38,313
that log base a of b minus log base a of c that this is equal to

27
00:01:38,313 --> 00:01:41,866
log base a of b over c.

28
00:01:41,866 --> 00:01:45,349
And this is really straight derived from these two right over here.

29
00:01:45,349 --> 00:01:48,066
Now with that out of the way let's see what we can apply.

30
00:01:48,066 --> 00:01:51,851
So right over here we have -- all of the logs are the same base --

31
00:01:51,851 --> 00:01:54,405
we have logarithm of x plus logarithm of 3,

32
00:01:54,405 --> 00:01:56,657
so by this property right over here --

33
00:01:56,657 --> 00:01:58,886
the sum of the logarithms of the same base

34
00:01:58,886 --> 00:02:08,824
this is going to be equal to log base 10 of 3 times x

35
00:02:08,824 --> 00:02:11,170
of 3x.

36
00:02:11,170 --> 00:02:14,049
Then based on this property right over here

37
00:02:14,049 --> 00:02:18,112
this thing can be rewritten, this is going to be equal to

38
00:02:18,112 --> 00:02:26,727
this can be written as log base 10 of 4 to the second power,

39
00:02:26,727 --> 00:02:30,396
which is really just 16.

40
00:02:30,396 --> 00:02:39,753
and then we still have minus logarithm base 10 of 2.

41
00:02:39,753 --> 00:02:42,099
And now using this last property --

42
00:02:42,099 --> 00:02:45,164
we know we have one logarithm minus another logarithm.

43
00:02:45,164 --> 00:02:50,504
This is going to be equal to log base 10 of 16 over 2,

44
00:02:50,504 --> 00:02:56,170
16 divided by 2, which is the same thing as 8.

45
00:02:56,170 --> 00:02:59,374
So right-hand side simplifies to log base 10 of 8

46
00:02:59,374 --> 00:03:04,297
the left-hand side is log base 10 of 3x.

47
00:03:04,297 --> 00:03:08,801
So if 10 to some power is going to be equal 3x,

48
00:03:08,801 --> 00:03:11,356
10 to the same power is going to be equal to 8.

49
00:03:11,356 --> 00:03:13,817
So 3x must be equal to 8.

50
00:03:13,817 --> 00:03:20,690
3x is equal to 8.

51
00:03:20,690 --> 00:03:26,077
And then we can divide both sides by 3.

52
00:03:26,077 --> 00:03:29,908
You get x is equal to 8 over 3.

53
00:03:29,908 --> 00:03:34,552
One way this little step here I said: look this is an exponent.

54
00:03:34,552 --> 00:03:37,107
If I raise 10 to this exponent I get 3x,

55
00:03:37,107 --> 00:03:39,057
10 to this exponent I get 8.

56
00:03:39,057 --> 00:03:41,309
So 8 and 3x must be the same thing.

57
00:03:41,309 --> 00:03:43,515
One other thing you could think about this is:

58
00:03:43,515 --> 00:03:46,650
let's take 10 to this power on both sides.

59
00:03:46,650 --> 00:03:53,175
So you could say 10 to this power and then 10 to this power over here.

60
00:03:53,175 --> 00:03:58,353
If I raise 10 to the power that I need to raise 10 to get 3x

61
00:03:58,353 --> 00:04:00,280
well I'm just going to get 3x!

62
00:04:00,280 --> 00:04:05,319
If I raise 10 to the power that I need to raise 10 to get 8

63
00:04:05,319 --> 00:04:07,455
I'm just going to get 8.

64
00:04:07,455 --> 00:04:09,243
So once again you get 3x is equal to 8.

65
00:04:09,243 --> 99:59:59,999
And you can simplify,you get x is equal to 8/3.