# Logarithm Rules (aka Log Laws) Explained with Examples

### What is Logarithm Rules (Definition)

To understand Logarithm Rules, you first need to know what is Logarithm or Log.

**What is Log or Logarithm?**

Logarithm is basically a power on any number that we use to get the other number defined in the Log.

**Examples:**

**Question: **What is the value of** log _{2 }32? Answer:** 5

Explanation: **2 ^{5 }**= 32

We need to get 32 by multiplying 2 with 2 and see how many times it goes. Sp 2 x 2 x 2 x 2 x 2 = 32. This will get us **2 ^{5}**

This is the reason the value of **log _{2 }32** = 5

**Question: **How about** log _{3 }81? Answer:** 4 (because

**3**)

^{4 }**Tricky Question: **Get the value of **Log 1000?**

**Answer:** 3 (But how?)

Explanation: When there is no base value associated with Log, we assume it’s **log _{10 }**

so **log _{10 }1000 = 10^{3}**

and the answer is 3.

**What are the Rules of Logarithm (Log Laws Table)**

Name of the Rule | Rule |

Product Rule of Logarithm | log_{a }xy = log_{a }x + log_{a }y |

Power Rule of Logarithm | log_{a }x = ^{y}y log_{a }x |

Quotient/Ratio Rule of Logarithm | log_{a }x / y = log_{a }x – log_{a }y |

Base Switch Rule of Logarithm | log_{a }b = 1 / log_{b }a |

Base Change Rule of Logarithm | log(_{a}x) = log(_{b}x) / log_{b }a |

Derivative of Logarithm | fx = log_{a }x ⇒ f ‘ (x) = 1 / ( x ln(a) ) |

Integral of Logarithm | ∫ log_{a }x dx = x ( log_{a }x – 1 / ln(a) ) + b |

Logarithm of 1 | log_{a }1 = 0 |

Logarithm of 0 | log_{a }0 is undefined |

Logarithm of the Base | log_{a }b = 1 |

Logarithms of Infinity | lim log_{a }x = ∞, when x→∞ |

Download the Log Table in** Image Format** or **PDF Format**

**1. Solved Examples for Product Rule of Logarithm**

**Rule: ** log_{a }*xy* = log_{a }*x** + *log_{a }*y*

**Question: **Solve this: **log _{2 }**

The same question can also be written as

**4*16**using Log Law.The same question can also be written as

**log**_{2 }4 + log_{2 }16**Answer: **

log_{2 }*4*16 *

*=> log _{2 }4 + log_{2 }16*

=> *log _{2 }2 ^{2} + log_{2 }2 ^{4}*

=> 2*log _{2 }2 + 4log_{2 }2*

=> 2 * 1 + 4 * 1

=> 2 + 4

=> 6

so ** **log_{2 }*4*16 = 6*

**2. Solved Example for Power Rule of Logarithm**

Rule: log_{a }*x ^{y}* =

*y*log

_{a }

*x*

**Question: **Solve this **log_{3}(3^{27})**

This question can also be written as **27log_{3}3**

**Note: **This type of questions are usually asked as objective questions because it doesn’t have much to do. Usually it is combined with other rules in question that we have also done after cover other rules. Scroll down if you are in rush to see.

**Answer:**

log* _{3}*(3

^{27})

=> 27log* _{3}*3

=> 27 * 1

=> 27

Answer is 27.

**3. Solved Example for Quotient/Ratio Rule of Logarithm**

**Rule: ** log_{a }*x / y* = log_{a }*x** – *log_{a }*y*

**Question: **Solve** log _{4 }1024 – log_{4 }16**

**Answer: **

log_{4 }*1024** – *log_{4 }*16 *

*=> log _{4 }(1024 / 16)*

=> *log _{4 }64*

=> log* _{4 }*4

^{3}

=> 3log* _{4 }*4

=>3 * 1

=> 3

S0, log_{4 }*1024** – *log_{4 }*16 = 3*

**4. Solved Example for Base Switch Rule**

**Rule:** log_{a }*b* = 1 / log_{b }*a*

**Question:** 1 / log_{2 }*128*

**Answer: **

1 / log_{2 }*128*

=> 1 / log_{2 }*2 ^{7}*

=> 1 / 7 * 1

=> 1 / 7

=> 0.1429

**Now let’s mix it up. Mix different types of Log Laws in one example.**

**5. Solved Example (Mixed of different Rules)**

**Question:** Solve this: **log _{3 }9 + log_{3 }81 – log_{5 }1250 + log_{5 }2**

**Answer:**

log_{3 }*9** + *log_{3 }*81 + log _{5 }1250 – log_{5 }2*

=> log_{3 }*(9 * 81)** + **log _{5 }(1250 / 2)*

=> log_{3 }*729** + **log _{5 }625*

=> *log _{3 }3^{6 }+ *

*log*

_{5 }5^{4}=> 6*log _{3 }3^{ }+ 4*

*log*

_{5 }5=> 6*1 + 4*1

=> 6+4

=> 10

So the value of log_{3 }*9** + *log_{3 }*81 + log _{5 }1250 – log_{5 }2 *= 10

**6. Solved Exmple: Expand this ***log*_{8 }(64_{k }4 / _{n}9)

*log*

_{8 }(64_{k }4 /_{n}9)*log _{8 }(64_{k }4 / _{n}9)*

=> *log _{8 }(64_{k }4) – log_{8 }n^{9}*

=> log_{8 }*64 *+ *log _{8 }k^{4} – log_{8 }n^{9}*

=> *log _{8 }8^{8 }*+ 4

*log*– 9

_{8 }k*log*

_{8 }n=> 8*log _{8 }8^{ }*+ 4

*log*– 9

_{8 }k*log*

_{8 }n=> 8 * 1* ^{ }*+ 4

*log*– 9

_{8 }k*log*

_{8 }n=> 8* ^{ }*+ 4

*log*– 9

_{8 }k*log*

_{8 }nSo, *log _{8 }(64_{k }4 / _{n}9) = 8^{ }+ 4log_{8 }k – 9log_{8 }n*

**Watch this video for examples and understanding of the Logarithm Law**

**Do you have other questions? You can type in the comment.**