If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Class 12 Physics (India)

### Course: Class 12 Physics (India)>Unit 13

Lesson 2: Logarithms

# Intro to logarithm properties

Learn about the properties of logarithms and how to use them to rewrite logarithmic expressions. For example, expand log₂(3a).
The product rulelog, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis
The quotient rulelog, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis
The power rulelog, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, dot, log, start base, b, end base, left parenthesis, M, right parenthesis
(These properties apply for any values of M, N, and b for which each logarithm is defined, which is M, N, is greater than, 0 and 0, is less than, b, does not equal, 1.)

#### What you should be familiar with before taking this lesson

You should know what logarithms are. If you don't, please check out our intro to logarithms.

#### What you will learn in this lesson

Logarithms, like exponents, have many helpful properties that can be used to simplify logarithmic expressions and solve logarithmic equations. This article explores three of those properties.
Let's take a look at each property individually.

## The product rule: $\log_b(MN)=\log_b(M)+\log_b(N)$log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis

This property says that the logarithm of a product is the sum of the logs of its factors.
We can use the product rule to rewrite logarithmic expressions.

### Example: Expanding logarithms using the product rule

For our purposes, expanding a logarithm means writing it as the sum of two logarithms or more.
Let's expand log, start base, 6, end base, left parenthesis, 5, y, right parenthesis.
Notice that the two factors of the argument of the logarithm are start color #11accd, 5, end color #11accd and start color #1fab54, y, end color #1fab54. We can directly apply the product rule to expand the log.
\begin{aligned} \log_6(\blueD5\greenD y)&=\log_6(\blueD5\cdot \greenD y) \\\\ &=\log_6(\blueD5)+\log_6(\greenD y)&&{\gray{\text{Product rule}}} \end{aligned}

### Example: Condensing logarithms using the product rule

For our purposes, compressing a sum of two or more logarithms means writing it as a single logarithm.
Let's condense log, start base, 3, end base, left parenthesis, 10, right parenthesis, plus, log, start base, 3, end base, left parenthesis, x, right parenthesis.
Since the two logarithms have the same base (base-3), we can apply the product rule in the reverse direction:
\begin{aligned} \log_3(\blueD{10})+\log_3(\greenD x)&=\log_3(\blueD{10}\cdot \greenD x)&&{\gray{\text{Product rule}}} \\\\ &=\log_3({10} x) \end{aligned}

### An important note

When we compress logarithmic expressions using the product rule, the bases of all the logarithms in the expression must be the same.
For example, we cannot use the product rule to simplify something like log, start base, 2, end base, left parenthesis, 8, right parenthesis, plus, log, start base, 3, end base, left parenthesis, y, right parenthesis.

1) Expand log, start base, 2, end base, left parenthesis, 3, a, right parenthesis.

2) Condense log, start base, 5, end base, left parenthesis, 2, y, right parenthesis, plus, log, start base, 5, end base, left parenthesis, 8, right parenthesis.

## The quotient rule: $\log_b\left(\dfrac{M}{N}\right)=\log_b(M)-\log_b(N)$log, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis

This property says that the log of a quotient is the difference of the logs of the dividend and the divisor.
Now let's use the quotient rule to rewrite logarithmic expressions.

### Example: Expanding logarithms using the quotient rule

Let's expand log, start base, 7, end base, left parenthesis, start fraction, a, divided by, 2, end fraction, right parenthesis, writing it as the difference of two logarithms by directly applying the quotient rule.
\begin{aligned} \log_7\left(\dfrac{\purpleC a}{\goldD 2}\right)&=\log_7(\purpleC a)-\log_7(\goldD 2) &{\gray{\text{Quotient rule}}} \end{aligned}

### Example: Condensing logarithms using the quotient rule

Let's condense log, start base, 4, end base, left parenthesis, x, cubed, right parenthesis, minus, log, start base, 4, end base, left parenthesis, y, right parenthesis.
Since the two logarithms have the same base (base-4), we can apply the quotient rule in the reverse direction:
\begin{aligned} \log_4(\purpleC{x^3})-\log_4(\goldD{y})&=\log_4\left(\dfrac{\purpleC{x^3}}{\goldD{y}}\right)&&{\gray{\text{Quotient rule}}} \end{aligned}

### An important note

When we compress logarithmic expressions using the quotient rule, the bases of all logarithms in the expression must be the same.
For example, we cannot use the quotient rule to simplify something like log, start base, 2, end base, left parenthesis, 8, right parenthesis, minus, log, start base, 3, end base, left parenthesis, y, right parenthesis.

3) Expand log, start base, b, end base, left parenthesis, start fraction, 4, divided by, c, end fraction, right parenthesis.

4) Condense log, left parenthesis, 3, z, right parenthesis, minus, log, left parenthesis, 8, right parenthesis.

## The power rule: $\log_b(M^p)=p\log_b(M)$log, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis

This property says that the log of a power is the exponent times the logarithm of the base of the power.
Now let's use the power rule to rewrite log expressions.

### Example: Expanding logarithms using the power rule

For our purposes in this section, expanding a single logarithm means writing it as a multiple of another logarithm.
Let's use the power rule to expand log, start base, 2, end base, left parenthesis, x, cubed, right parenthesis.
\begin{aligned} \log_2\left(x^\maroonC3\right)&=\maroonC3\cdot \log_2(x)&&{\gray{\text{Power rule}}} \\\\ &=3\log_2(x) \end{aligned}

### Example: Condensing logarithms using the power rule

For our purposes in this section, condensing a multiple of a logarithm means writing it as another single logarithm.
Let's use the power rule to condense 4, log, start base, 5, end base, left parenthesis, 2, right parenthesis,
When we condense a logarithmic expression using the power rule, we make any multipliers into powers.
\begin{aligned} \maroonC4\log_5(2)&=\log_5\left(2^\maroonC 4\right)&&{\gray{\text{Power rule}}} \\\\ &=\log_5(16) \end{aligned}

5) Expand log, start base, 7, end base, left parenthesis, x, start superscript, 5, end superscript, right parenthesis.

6) Condense 6, natural log, left parenthesis, y, right parenthesis.

## Challenge problems

To solve these next problems, you will have to apply several properties in each case. Give it a try!
7) Which of the following is equivalent to log, start base, b, end base, left parenthesis, start fraction, 2, x, cubed, divided by, 5, end fraction, right parenthesis?