Logarithm – SS1 Mathematics Lesson Note (First Term)

The logarithm of a number is the power to which a base must be raised to give that number.

• Index form: y = n^x.
• Logarithmic form: x = log_ny.
• Example: log₁₀1000 = 3 because 10³ = 1000.

Behavioural objectives

By the end of the lesson, students should be able to:

1. change numbers from index form to logarithmic form and back;
2. find the logarithm of a number to base ten using tables;
3. state the integer part of the logarithm of a number by counting digits;
4. find the antilogarithm of a given logarithm using tables;
5. multiply and divide numbers using logarithm tables.

What is a logarithm?

A logarithm is the power to which a base must be raised to give a number, so if y = n^x, then x = log_ny.

• Base: the number n that is raised to a power.
• Logarithm: the power x, which is the answer when we write log_ny.
• Base ten logarithm: the power to which 10 is raised to give a number. If y = 10^x, then x = log₁₀y.
• Example: log₁₀1000 = 3 since 10³ = 1000.
• Example: log₁₀100 = 2 since 10² = 100.

How do we write numbers in logarithmic form?

To write an index statement in logarithmic form, make the power the answer and keep the base as the small number below “log”.

1. Identify the base (the number being raised to a power).
2. Identify the power (the index).
3. Write log, then the base as a subscript, then the number that the power gives.
4. Put an equals sign and the power.

Example 1. Express the following in logarithmic form.

1. 2^-6 = 1/64
2. 3⁵ = 243
3. 5³ = 125
4. 10⁴ = 10,000

Solutions

1. 2^-6 = 1/64, so log₂(1/64) = -6
2. 3⁵ = 243, so log₃243 = 5
3. 5³ = 125, so log₅125 = 3
4. 10⁴ = 10,000, so log₁₀10000 = 4

How do we write logarithms in index form?

To write a logarithm in index form, raise the base to the value of the logarithm and set it equal to the number.

1. Read the base from the subscript.
2. Read the value after the equals sign as the power.
3. Write base^power = the number inside the log.

Example 2. Express the following in index form.

1. log₂(1/8) = -3
2. log₁₀(1/100) = -2
3. log₄64 = 3
4. log₅625 = 4
5. log₁₀1000 = 3

Solutions

1. 2^-3 = 1/8
2. 10^-2 = 1/100
3. 4³ = 64
4. 5⁴ = 625
5. 10³ = 1000

How do we find logarithms to base ten from tables?

The logarithm of a number to base ten is found by writing the number in standard form and reading the logarithm of its first part from the table.

1. Write the number in standard form, A × 10ⁿ, where A is between 1 and 10.
2. Read the logarithm of A from the table. This is the decimal part.
3. Add the index n. This is the integer part.

Example 3. Use the tables to find the log of (a) 37 and (b) 3900 to base ten.

Solutions

(a) 37 = 3.7 × 10¹ (standard form)
= 10^0.5682 × 10¹ (from table)
= 10^{0.5682 + 1}
= 10^1.5682
Hence log₁₀37 = 1.5682

(b) 3900 = 3.9 × 10³ (standard form)
= 10^0.5911 × 10³ (from table)
= 10^{0.5911 + 3}
= 10^3.5911
Therefore log₁₀3900 = 3.5911

What are the integer part and the decimal part of a logarithm?

The integer part of a logarithm is the whole number before the decimal point, and the decimal part (the mantissa) is the fraction after it.

• Integer part: the whole number before the point. In log₁₀3900 = 3.5911, it is 3.
• Decimal part (mantissa): the fraction after the point. In 3.5911, it is .5911.
• Rule for numbers greater than 1: count the digits to the left of the decimal point, then subtract 1.
• Decimal part: always read from the tables.

Example 4. Use the logarithm table to find the logarithms to base ten of 51.38, 840.3 and 65160.

Solutions

What is the logarithm of a number less than 1?

The logarithm of a number between 0 and 1 has a negative integer part.

• Rule: count the zeros straight after the decimal point, add 1, and make the result negative.
• Example: 0.0039 = 3.9 × 10^-3, so log₁₀0.0039 = -3 + 0.5911.
• Bar notation: -3 + 0.5911 is written as bar 3 point 5911, with the minus sign over the 3 only.
• Example: 0.000352 = 3.52 × 10^-4, so the integer part of its logarithm is -4.

What is an antilogarithm?

An antilogarithm is the number whose logarithm is given, so it reverses the finding of a logarithm.

• Use of tables: the logarithm table can be used in reverse, but the antilogarithm table is more convenient.
• Decimal part: look this up in the antilogarithm table.
• Integer part: use this to place the decimal point.

1. Look up the decimal part of the logarithm in the antilogarithm table.
2. Read off the digits, written as a number between 1 and 10.
3. Take the integer part n of the logarithm.
4. Move the decimal point n places to the right (the answer has n + 1 digits before the point).

Example 5. Find the antilog of the following logarithms: 0.5682, 2.7547 and 5.3914.

Solutions

What are more examples of logarithms and antilogarithms?

The antilogarithm of the logarithm of a number gives back the original number.

How do we multiply and divide using logarithm tables?

To multiply with logarithms, add the logarithms of the numbers; to divide, subtract them.

• Multiplication law: log(a × b) = log a + log b.
• Division law: log(a ÷ b) = log a – log b.
• Power law: log(aⁿ) = n × log a.

Steps for multiplication

1. Find the logarithm of each number.
2. Add the logarithms.
3. Find the antilogarithm of the sum.

Steps for division

1. Find the logarithm of each number.
2. Subtract the logarithm of the divisor from the logarithm of the dividend.
3. Find the antilogarithm of the difference.

Example 6. Evaluate the following using tables.

1. 4627 × 29.3
2. 819.8 ÷ 3.905
3. (48.63 × 8.53) ÷ (15.39 × 3.52)

Solutions

(a) 4627 × 29.3

The last row is the sum: 3.6653 + 1.4669 = 5.1322. Therefore 4627 × 29.3 = 135600 (4 s.f.).

(b) 819.8 ÷ 3.905

The last row is the difference: 2.9137 – 0.5916 = 2.3221. Therefore 819.8 ÷ 3.905 = 209.9.

(c) (48.63 × 8.53) ÷ (15.39 × 3.52)

The last row is the numerator log minus the denominator log: 2.6178 – 1.7337 = 0.8841. Therefore (48.63 × 8.53) ÷ (15.39 × 3.52) = 7.658.

Evaluation

1. Find the log of (i) 0.009321 (ii) 0.5454.
2. Find the antilog of: (i) 3.3210 (ii) 1.8113 (iii) 0.5813 (iv) 3.2212.
3. Use logarithm tables to calculate: (a) 36.12 × 750.9 (b) 3577 × 31.11 (c) 256.5 ÷ 6.45 (d) 113.2 × 9.98.

Assignment

Answer all the following questions using logarithm tables where needed.

1. Find the log of 802 to base 10. (a) 2.9042 (b) 3.9040 (c) 8.020 (d) 1.9042
2. Find the number whose logarithm is 2.8321. (a) 6719.2 (b) 679.4 (c) 0.4620 (d) 67.92
3. What is the integer part of the log of 0.000352? (a) -4 (b) -3 (c) 4 (d) 3
4. Given that log₂(1/64) = m, what is m? (a) -5 (b) -4 (c) -6 (d) 3
5. Express the log in index form: log₁₀10000 = 4. (a) 10³ = 10000 (b) 10^-4 = 10000 (c) 10⁴ = 10000 (d) 10⁵ = 100000
6. Evaluate using logarithm table 6.28 × 304 and express your answer in the form A × 10ⁿ, where A is a number between 1 and 10 and n is an integer.
7. Use the logarithm table to calculate 6354 × 6.243 correctly to 3 s.f.