00:00In the name of Allah, the most Gracious, the most Merciful.
00:13Peace be upon you, dear students and viewers.
00:15In today's video, we are going to do the exercise 3.4 of our logarithm chapter.
00:23Before starting the exercise, we are going to do an important example.
00:29After that, we will see the complete exercise 3.4.
00:32Applications of the Laws of Logarithm in Numerical Calculation
00:38This is an example.
00:39a is equal to a base 0 e raised to the power minus kd.
00:44If k is equal to 2, then what should be the value of 1?
00:49You can say that this is the level of the equation 1.
00:54a is equal to a base 0 divided by 2.
01:00Put the value of k and a in 1.
01:06a base 0 divided by 2 is equal to a0 e raised to the power minus 2d.
01:14Because k value is 2.
01:17Now, 1 divided by 2 is equal to e raised to the power minus 2d.
01:28This a0 is considered to be a0 available on the left hand side of the equation.
01:35Now, taking common log of both sides.
01:37Common log means the log which has base 10, this one.
01:41Log 10 1 by 2 is equal to log base 10 2 e raised to the power minus 2d.
01:46Applying quotient rule and power rule.
01:50Log base 10 to 1 minus log base 10 to 2 is equal to minus 2d.
01:58This power becomes coefficient or multiplier of the log base 10 to e.
02:050 minus 0.0301.
02:09Log base 10 to 1 is 0.
02:14And log base 10 to 2 is equal to 0.301 is equal to minus 2d.
02:20Log base 10 to e is equal to 2.718.
02:290.301 is equal to minus 2d into.
02:35Here, I have put the value of e which is equal to 2.718.
02:38Now, I am going to check these values into the log table to get this value 0.4343.
02:48So, dividing both sides by 0.434 into 2.
02:59We get d is equal to 0.3466.
03:03This minus is cancelled out by this minus.
03:10Let me correct it here.
03:12So, this is our required answer.
03:14Now, we are going to start exercise 3.4.
03:17The first question is here.
03:19Use the log table to find the value of 0.8176 into this x means multiplication here 13.64 solution.
03:31Let we give a name to this numerical expression which is y is equal to 0.8176 into 13.64.
03:44Now, taking log on both sides.
03:46Log y is equal to log 0.8176 into 13.64 as the numerical expression is in the form of product.
03:56So, we are going to apply the product laws of logarithm here.
04:00Log y is equal to log 0.8176 plus log 13.64.
04:05You have noticed that in log product the multiplication sign turns into the positive sign.
04:12Now, we are going to find the statistics of the two given numbers.
04:180.8176 is minus 1 because we start from point and there is no number is before point.
04:29There is 0.
04:30So, we are going to put minus 1 here.
04:32And how many digits are there before point 3, 1 or 2.
04:38So, we will subtract 1 from 2.
04:40We will get statistics of 1 for this 13.64.
04:46Mantissa, consider only 4 significant digits.
04:510.8176, 13.64, this one, 9.122 plus 3 and 13.35 plus 13.
05:06And by adding this one, we will get this.
05:10Log y is equal to, this is the log value of these two given numbers.
05:17So, minus 1.9125 plus 1.1348.
05:24When you will visit the log table, not antilog table, log table you will see this.
05:31Number 81 against the same horizontal line against the 7.
05:37And then again in the second, third part of the log table under 6.
05:44So, this is the way we have got the log values of these two numbers.
05:50And now we are going to put minus 1.
05:53Minus 1, it means statistics is minus 1.
05:58So, we read it as bar 1.9125 plus 1.1348.
06:04So, I am going to be split into this one, minus 1 plus 0.9125 plus 1 plus 0.1348.
06:13And we will get log y is equal to 1.0473.
06:22Taking antilog of both sides, look at this antilog is cancelled out by this log and only y is remaining.
06:30And now we are going to move to the antilog tables to find the values of 0.0473.
06:37This is our mantissa.
06:39Always remember that when you are going to use antilog table, you only focus on the mantissa.
06:45Because this first digit before point stands for statistics here.
06:52So, currently ignore it.
06:56And now look only decimal portion into the antilog tables that 0.0473.
07:011114 plus 1 is equal to 1115.
07:07As statistics is 1, so we are going to add 1 into this one.
07:12Therefore, add 1 to get position of decimal which is 2.
07:17So, we are move from left to right 1, 2.
07:22After that we will place the point here.
07:26And this is our required answer.
07:28Now we are going to move to the second part of the first question which is 789.5 raised to power 1 by 8.
07:36This is our exponent is involved here.
07:39Let y is equal to 789.5 raised to power 1 by 8.
07:43Taking log of both sides.
07:45Let log y is equal to log 789.5 raised to power 1 by 8.
07:55As this power becomes coefficient of the log number.
07:59So, under the rule of...
08:02And this is the exponent rule of logarithms.
08:051 by 8 into log 789.5.
08:10Statistics is here.
08:11Look at this how many digits are there.
08:13Before point, there are 3 digits.
08:16So, we are going to subtract 1 to get the statistics of this number.
08:20So, we have statistics 3 minus 1 is equal to 2.
08:23And this has been determined by the inspection method.
08:27So, mantissa is equal to 789.5.
08:31We consider 4 significant digits.
08:33Log 78 under column 9 and 5.
08:361 by 8, 8971 plus 3.
08:412.8971 is equal to 0.3621 also divided by 8.
08:46Now, we are going to check this 0.3621 into the table.
08:53So, we will get this 2302 plus log y is equal to 0.3621.
08:59Here, statistics is 1.
09:02Taking antilog of both sides.
09:04Antilog y is equal to antilog 0.3621.
09:092301 plus 1 and 2302.
09:12So, answer is 2.302.
09:20Now, we are going to move to the part 3 of the question number 1.
09:28Here is the part 3.
09:300.678 into 9.01 divided by 0.0234.
09:37Now, look at this.
09:39We have product and quotient in this part.
09:43So, we are dealing with the product and quotient rules of logarithm in this question.
09:50So, let y is equal to 0.678 into 9.01 divided by 0.0234.
09:57Taking log of both sides.
09:59Log y is equal to log.
10:01By applying product and quotient rule of logarithm, we will get log 0.0678.
10:07Plus this multiplication sign turns into positive.
10:11Log into 9.01 divided by log 0.0234.
10:17If you apply the quotient rule here, you can write it like this.
10:22Minus log 0.0234 as well.
10:26Statistics is 1 bar 1.
10:30And for the 9.01, the statistics is 0.
10:33Because for decimal, there is only 1 digit.
10:361 minus 1 is equal to 0.
10:38And for 0.0234, the statistics is bar 2.
10:45Means how many zeros are offered after decimal.
10:52Only one zero and two digits.
10:54So, 1 and 2.
10:57So, statistics is bar 2.
10:59Now, we are going to deal with the mantissa.
11:020.0678 is 8312.
11:05You will get this value from the log tables.
11:08You will see 67 in the same horizontal line under the column 8.
11:13So, you will get this 8312 value.
11:16And log value is 1 bar minus 8312.
11:20For 9.01 is 9547.
11:24Here you look 90.
11:27Value of 90 and guess.
11:31Column 1, under the same horizontal line, you will get 9547.
11:35And log value is 0.9547.
11:39Because statistics is 0.
11:41For 0.0234 is 3692.
11:45And the log value is 2 bar 2.3692.
11:49Thus, log y is equal to bar 1 minus 0.8312 plus 0.9547 minus.
11:58Because it is available in the denominator part of the equation.
12:05So, it will turn into the negative sign.
12:08If you are going to use the quotient laws of logarithm.
12:13Now, log y is equal to 1 bar.
12:16Bar 1 plus 0.8312.
12:19Split this one.
12:200.09547.
12:22Thus, again split this one.
12:24This minus and this minus turns into positive 2 minus.
12:27This plus into minus turns into negative 0.3692.
12:33Log y is equal to 1 plus.
12:37By adding this one and this one.
12:391.4167 and 2.4167.
12:43Taking antilog of both sides.
12:45We will get y is equal to 251.
12:49And you will check this.
12:520.41.
12:54Guess 6.
12:56Column 6 and again column 7.
12:58You will get this value 251.
13:00This is our answer.
13:02Now, we are move to the equation number 4.
13:05Part 4 of the equation number 1.
13:07Under 5th square root 2.709.
13:13Into 7th square root 1.239.
13:17Again, 2 rules are involved in this equation.
13:23Disease rule and product rule.
13:27Let y is equal to 5th square root 2.709.
13:31Into 7th square root 1.239.
13:34Is equal to 2.709 raised to the power of 1 by 5.
13:39Into 1.239 raised to the power of 1 by 7.
13:44What I have done here.
13:45I have removed the square root sign.
13:47And turned it to the power.
13:51Taking log of both sides.
13:52Log y is equal to log 2.709 raised to the power of 1 by 5.
13:56Into 1.239 raised to the power of 1 by 7.
13:59Applying product law of logarithm.
14:07Log y is equal to log 2.709 raised to the power of 1 by 5.
14:10Plus log 1.239 raised to the power of 1 by 7.
14:13Now, applying disease rule of logarithms.
14:16This power becomes coefficient of this log number.
14:191 by 5 log 2.709.
14:22Plus 1 by 7 into log 1.239.
14:27Using log tables.
14:29Getting the log values of the numbers.
14:31Log 1 by 5 is equal to 4.314 plus 1.4.
14:35Plus 1 by 7 into 0.0899 plus 32.
14:41You can use decimal as well here.
14:45But both places.
14:470.4314 plus 0.0014.
14:51So, we will get 0.4328.
14:54And 0.0931.
14:56Keeping the coefficient same.
14:59Now, dividing this element by 5.
15:02And this element by 7.
15:04We will get 0.0856.
15:06Plus 0.13286.
15:10By adding up them.
15:11We will get 0.0999999.
15:14Taking antilog of both sides.
15:16Log there.
15:17Statistic is 0.
15:19So, we will turn into the 1.
15:21By increasing its value by 1.
15:230 plus 1 is equal to 1.
15:25So, 1, 2, 5, 6.
15:27When you look.
15:280.09.
15:29And guess 9.
15:30Under column 9.
15:31And then again 9.
15:33You will get 1, 2, 5, 9.
15:35Statistic is 1.
15:36So, we will move from left to right.
15:38One digit.
15:391.259 to place decimal.
15:42This is our answer.
15:44So, remaining two parts.
15:47And the other questions.
15:48We will do in the next video.
15:50I hope you will find this video useful.
15:52If you find useful.
15:54Please hit the like button.
15:55And share with your friends.
15:56Thanks for watching.
15:57Assalam-o-Alaikum.
