Lesson
Title: Logarithms of Numbers.
Objectives :
At the end of this lesson, you will be able to,
1. Prove the concept of logarithms.
2. Solve WASSCE and other written examination questions on logarithms.
3. Solve JAMB questions on logarithms.
What you should know :
You should be familiar with indices.
Introduction :
In this lesson, you will use a scientific calculator instead of the logarithm table due to examination time constraints.
Before the full acceptance of scientific calculators into examination halls, logarithm tables were used for finding the logarithm and antilogarithms of an expression, but today, there's a provision for branded scientific calculators to be used in examination halls.
Presentation :
Concept of logarithms: In indices, 10000 = 104 to express this in logarithm, 10 is the logarithm base of 10000 = 4 and this simply means, log10 10000 = 4.
The logarithm of a number is the power that is raised to give that number. In other words, logarithm is another word for power although logarithms can be in any other base than 10.
E.g 81 = 34 in logarithm log381= 4.
Generally, if y = nx , then x = logny. In base 10
If y = 10x then x = log10y.
Multiplication and division of logarithms.
Calculation using logarithms is similar to the laws of indices since they have the same base which is similar to the power of 10.
N.B Multiplication and division laws of indices mean addition, and subtraction of the powers respectively, the same laws apply to logarithms.
Examples (i) 4.682 x 7.245 (ii) 4.562 ÷ 98.76.
Step 1: we will solve the above problems for WASSCE and JAMB styles.
For WASSCE, study the image below
In WASSCE and other written examinations, you're required to show your workings. From the above image, the left column (No) contains the numbers while the right column (log) contains the logarithm value for the 1st and 2nd numbers.
=> Using your calculator, press the following, Iog 4.682 which is 0.6704, and log 7.245 = 0.8600. N.B. always leave your logarithm value in 4 decimal places.
=> add the values, 0.6704 and 0.8600 = 1.5304 N.B multiplication in logarithm means addition.
=> Write the result as shown above.
=> Now convert log 1.5304 to a number, this is done by pressing "SHIFT LOG 1.5304 =" on your calculator which is 33.91 and it should be written in the "No" column.
=> Therefore, 4.682 x 7.245 = 33.91.
For JAMB and QUIZ scenarios:
=>Remember multiplication means addition.
=> Press the following on your calculator,
log 4.682 + log 7.245 which is 1.5304.
=> Write the answer out or press the following while the answer is still showing, "SHIFT LOG =" or "SHIFT LOG 1.5304 =" The result is the anti-log of 1.5304 = 33.92.
The above steps are applicable when you want to solve 4.562 ÷ 98.76, only change the signs. N.B ÷ means -
Example 3
Given that log10 2 = 0.3010, log10 7 = 0.8451 and log10 5 = 0.6990. Evaluate the following without using logarithm table i. Log1035. ii. Log102.8 (WASSCE).
Step i :
=> This is a written exam and you need to show your workings.
=> N.B 2.8 = 28/10
=> Find the factors of 35 and 28/10 for the given logarithm values.
=> 35 = 7 x 5
=> 28/10 = 4*7/10 = 2² * 7/10.
=> log10 7 x 5 where log10 7 = 0.8451
and log105 = 0.6990.
=> log10 0.8451 + 0.6990 = 1.5441
=> Therefore, log1035 = 1.5441
Step ii.
=> log10 2.8 is equivalent to log10 2² x 7/10.
=> log10 2² x log10 7 ÷ log1010.
=> From the given logarithm values, 2 = 0.3010 and 7 = 0.8451. N.B log1010 = 1.
=> 0.3010 x 2 + 0.8451 - 1 = 0.4471.
N.B Product rule of indices was applied to 2² = 2x2, i.e. the logarithm value of 2 x 2. In logarithm, multiplication (x) = addition (+) while division (÷) = subtraction (-).
=> Therefore, log10 2.8 = 0.4471.
Example 4. log520 = x. Find x (JAMB)
=> From the general rule, log5 20 = x is equivalent to log 20 = log5x.
=> log5x = xlog5 (product rule)
=>log20 = xlog5.
=> Find the logarithms of both sides.
=> Press log 20 and log 5 on your calculator to get the values.
=> 1.3010 = x0.6990
=> Divide both sides by 0.6990 to get x.
=> 1.3010/0.6990 = x0.6990/0.6990.
=> 1.8612 = x.
=> Therefore, x = 1.8612.
Summary / Conclusion:
=> Logarithms is another word for power, i.e. if x = ab then logax = b.
=> When solving logarithm problems, ensure you write the solution steps sequentially.
=> Your logarithm values should be in 4 decimal places unless stated otherwise.
=> When asked to solve an expression using a logarithm, ensure you convert the expression result back to a number using antilog.
Exercise :
i. Using logarithm, solve 4.562 ÷ 98.76
ii. Given log 5 = 0.6990, evaluate without using logarithm table log 2.5