### Video Transcript

Given that log three to the base seven is approximately equal to 0.5646, find, without using a calculator, the value of log 147 to the base seven correct to four decimal places.

Before starting this question, we need to recall one of our laws of logarithms: log π₯ plus log π¦ is equal to log π₯π¦. 147 is equal to three multiplied by 49. This means that log of 147 to the base seven can be rewritten as log of three multiplied by 49 to the base seven. Using the law that we have quoted, this in turn can be rewritten as log three to the base seven plus log 49 to the base seven.

Weβre told in the question that log three to the base seven is approximately equal to 0.5646. 49 is equal to seven squared. Therefore, log 49 to the base seven can be rewritten as log of seven squared to the base seven.

At this point, we need to consider another one of the laws of logarithms: log π₯ to the base π is equal to π log π₯. We can, therefore, rewrite log of seven squared as two log seven. Log of π to the base π is equal to one. This means that log of seven to the base seven will also be equal to one. Our expression, therefore, simplifies to 0.5646 plus two multiplied by one. As two multiplied by one is two, weβre left with 2.5646.

The value of log 147 to the base seven correct to four decimal places is 2.5646.