Properties of Logarithms

Properties of Logarithms
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Astronomy involves calculations with large numbers such as:

\[ \begin{align} (4.2394 \times 10^{27})^{2391}\\ 2897 \times 857^{25}\\ (0.00054316)^{43}\end{align}\]

In such cases, logarithms are helpful.

It does not involve the use of calculators.

A telescope with Planets and stars in the background

In this mini-lesson, we shall explore the world of logarithms by finding answers to questions like what is the meaning of log base e, what are the basic properties of logarithms, and how to find logarithms using log table.

Lesson Plan

What Do You Mean by log base e?

Before we begin, let us understand the log definition.

An exponential form can be converted to logarithmic form.  \[ \begin{align} a^x &= N \\ \log_a N& = x\end{align}\]

In the above expression, the base of the logarithm is \(a\).

\[ \begin{align} 2^5 &= 32 \\ \log_2 32& = 5 \end{align}\]
Here the base of the logarithm is 2.

Considering the different bases, logarithms are named as common logarithms and natural logarithms.

Natural Logarithms

As per log definition, the logarithms to the base \(e\) are called natural logarithms. \[ \begin{align} \log_e2 &= 0.693147 \\ e& =2.718281 \end{align}\]

Logarithms are also expressed to the base 10. 

The logarithms to the base of 10 are called common logarithms.  \[  \log_{10}2 = 0.3010\]


What Are the Three Basic Properties of Logarithms?

The properties of logarithms include the following three basic log formula.

Product Rule:  \[ \log xy = \log x + \log y\]

Division Rule:  \[ \log \frac{x}{y} = \log x - \log y\]

Exponential Rule: \[ \log x^m= m\log x \]


Log Table

The log table to the base of 10 is used to find the logarithm of numbers.

These logarithms are called as common logarithms.

Let us try to understand the process of finding logarithms with the help of an example.

Find log 32.57

STEP - 1:  Look for row 32 and column 5. You will obtain the value 5119

STEP - 2: In the same row 32, look for the mean difference under column 7. The mean difference is 9.

STEP - 3:  Mantissa: It is the sum of the above two values of 5151 and 9.  \[ 5119 + 9 = 5128\]

STEP - 4:  Characteristic: This is found from the integral part ie 32.  For a two-digit number, the characteristic is 1.

STEP - 5:  Hence the final value is \(\log 32.57 = 1.5128 \)

Logarithm Table

 
important notes to remember
Important Notes
  1. The exponential form \(a^x = N \) can be converted to logarithmic form as  \(log_aN = x \)
  2. \[ \log 1 = 0  ~and~ \log_a a = 1\]
  3. Product Rule:  \[ \log ab = \log a + \log b\]

  4. Division Rule:  \[ \log \frac{a}{b} = \log a - \log b\]

  5. Exponential Rule: \[ \log a^m= m\log a \]

  6. \[ \log_b a = \dfrac{\log a}{ \log b}\]
  7. \[ \log_b^{n}a^m =\dfrac{m}{n} \log_ba\]
  8. A number raised to a logarithm power with the same base is \( a^{\log_ax} = x\).

How to Use Logarithm Properties?

The Logarithm properties make calculations easier.

  • Logarithms are used to convert exponential form, \(2^5 = 32  \) into logarithmic form \(log_2 32 = 5 \).
  • The logarithm property is used to write a product as a sum. \[ \log 14 = \log (7 \times 2) = \log 7 + \log 2\] 
  • The logarithm property is used to write a division as a difference.  \[ \log 0.3 = \log {\frac{3}{10}} = \log 3 - \log 10\]
  • The logarithm property is used to write an exponent as a product.  \[\log \sqrt 5 = \log 5^{\frac{1}{2}} = \frac{1}{2} \log5 \]
  • The logarithm property is used to split a large number into its smaller factors.  \[ \log24 = \log (8 \times 3) = \log(2^3 \times 3) = \log2^3 + log3 = 3log2 + log3\]

Solved Examples

Example 1

 

 

Express \(2^5 = 32 \) in logarithmic form.

Solution

The exponential form \(a^x = N \) can be written in logarithmic form as \(\log_aN = x \).

Hence, \(2^5 = 32 \) can be written in logarithmic form as \(\log_2{32} = 5\)

\(\therefore \)The logarithmic form is \(\log_2{32} = 5\).
Example 2

 

 

Given \(\log 5 = a \). 

What is the value of \(\log \frac{1}{125} \) ?

Solution

\(\log 5 = a \) ...... (Given)
\[\begin{align} log \frac{1}{125} &= log \frac{1}{5^3} \\&=log 5^{-3}\\&=-3log 5\\&=-3a\end{align}\]

\(\therefore \) The answer is -3a.
Example 3

 

 

Find the value of \(\log 36 \). 

We have \(\log 2 = 0.3010 \) and \( \log 3 = 0.4771 \).

Solution

Given \(\log 2 = 0.3010 \) and \( \log 3 = 0.4771 \)
\[\begin{align}\log 36 &= \log (2 \times 2 \times 3 \times 3) \\&=\log(2^2 \times 3^2) \\&=\log2^2 + \log 3^2 \\&= 2\log 2 + 2\log 3\\&=2(0.3010) + 2(0.4771) \\&= 0.6020 + 0.9542 \\&=1.5562 \end{align} \]

\(\therefore \) \(\log36 = 1.5562 \)
Example 4

 

 

Solve \(16^{\log_45}\).

Solution

Here we can use the formula \( a^{\log_ax} = x\).
\[\begin{align}16^{\log_45}& = 4^{2\log_45}\\& =4^{\log_45^2} \\&=5^2 \\&= 25\end{align}\]

\(\therefore \) The answer is 25.
 
Challenge your math skills
Challenging Question
  1. Find the value of \(\log 1200 \)
  2. Find the value of \(\log 15 \)
  3. Find the value of \(\log 0.0009 \)

(Given \(\log 2 = 0.3010 \) and \(\log 3 = 0.4771 \) )

Interactive Questions on Properties of Logarithms

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of properties of logarithms. The math journey around properties of logarithms starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Properties of Logarithms

1. What are logarithms used for?

Logarithms are used to make calculations easier.

The calculations of multiplication and division are changed to addition and subtraction with the help of logarithms.
\[ \begin{align} \log ab &= \log a + \log b \\ \log \frac{a}{b} &= \log a - \log b\end{align}\]

2. What is a logarithm in simple terms?

Logarithms make complex calculations easier.

The exponential form can be converted into logarithmic form. 
\[ \begin{align} a^x &= N \\ \log_a N& = x\end{align}\]

3. What is log 10 equal to?

The value of log10 is:\[\log_{10}10 = 1 \]

4. What is log 5 equal to?

The value of \log5 is: \[\log5 = 0.6989 \]

5. What do you mean by log e?

The meaning of log e is log to the base of e..\[ \log_e e = 1\] 

The log to the base e is called a natural logarithm.

6. Can the base of the log be negative?

The base of a logarithm cannot be a negative number.

7. What is the value of log1?

The value of log1 is 0.  \[ \log_{any base}1 = 0\] 

7. How many laws of logarithms are there?

There are five laws of logarithms:

  1. Product Rule:  \( \log ab = \log a + \log b\)
  2. Division Rule:  \( \log \frac{a}{b} = \log a - \log b\)
  3. Exponential Rule: \( \log a^m= m\log a \)
  4. \( \log_b a = \dfrac{\log a}{ \log b}\)
  5. \( a^{\log_ax} = x\)

8. What is the first law of logarithm?

The first law of logarithms is used to convert the product of two numbers into addition.  \[ \log ab = \log a + \log b\] 

9. How many types of logarithms are there?

There are two types of logarithms.

Natural Logarithms: These are logarithms to the base of e.  e = 2.7182
Common Logarithms:  These are logarithms to the base of 10.

10. What are the three properties of logarithms?

The three properties of logarithms are the following three log formula.

  1. \( \log ab = \log a + \log b\)
  2. \( \log \frac{a}{b} = \log a - \log b\)
  3. \( \log a^m= m\log a \)
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