Logarithms: Multiplication and Division, Log Tables

• Logarithms of Numbers to Base 10.
• Multiplication and Division of Numbers Using Logarithms Tables.

LOGARITHMS OF NUMBERS TO BASE 10

In general the logarithm of a number is the power to which the base must be raised in order to give that number. i.e if y=n^x, then x = log_ny. Thus, logarithms of a number to base ten is the power to which 10 is raised in order to give that number i.e if y =10^x, then x =log₁₀y. With this definition log₁₀1000 = 3 since 10³= 1000 and log₁₀100 = 2 since 10²=100.

1. Express the following in index form
2. Log₂(1/8) = -3      (b)  Log₁₀(1/100) = -2      (c) Log₄64 = 3      (d)  Log₅625 = 4    (e) Log₁₀1000 = 3

Solutions

1. Log₂ (1/8)= -3

Then 2^-3 = 1/8

• Log₁₀(1/100) = -2

Then 10^-2 = 1/100

• Log₄64 = 3

Then 4³ = 64

• Log₅625 = 4

Then 5⁴ = 625

• Log₁₀1000 = 3

Then 10³ = 1000

Note: Logarithms of numbers to base ten are found with the help of tables

How to use Log Table

Examples:

Use the tables to find the log of:

• 37     (b) 3900 to base ten

Solutions

1. 37 = 3.7 X 10

=3.7 X 10¹(standard form)

=10^{0.5682 + 1} X10¹ (from table)

=10^1.5682

Hence log₁₀37 = 1.5682

• 3900 = 3.9 X1000

=3.9 X 10³ (standard form)

=10^0.5911 X 10³ (from table)

=10^{0.5911 + 3}

=10^3.5911

Therefore log₁₀3900 = 3.5911

Antilogarithms table

Antilogarithm is the opposite of logarithms. To find number whole logarithm is given. It is possible to use logarithm table in reverse

However, it’s convenient to use the tables of antilogarithms. When finding an antilogarithm, look up the fractional part only, then used the integer to place correct the decimal point correctly in the final number

READING ASSIGNMENT

New General mathematics SSS1, page 21, Exercise 1h 1 – 3.

WEEKEND ASSIGNMENT

1. Find the log of 802 to base 10 (use log tables) (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 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