In Exercises 1–40, use properties of logarithms to expand each lo...

Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.log7 (7x)

Accepted Answer

Hello. Today we're going to be expanding the given logarithms using the properties of logarithms. So the logarithms given to us is the logo rhythm of base of 13 times X. Now on the inside argument of our given logarithms, the insane argument is evolving multiplication. So that's an indication that we're going to need to use the multiplication property of logarithms to expand the given logarithms. The multiplication property states that if you have the log base of A. Of argument B plus log base A. Of argument. See then you can go ahead and combine these logarithms given that the bases are the same as a single logarithms as log base A. Of B times C. So we're gonna be using this multiplication property but we're gonna be going from right to left because the given logarithms already has a product inside its argument. So we're gonna want to be separating that into a sum of logarithms with the same base. So let's go ahead and take a look at the two numbers that are being multiplied. The two numbers are 13 and X. So using our multiplication property we can expand this by writing it as logarithms of base 13 of 13 plus logarithms of base 13 of X. Now there is one more simplification simplification, we can do remember that if your logo rhythm has a base that matches the inside argument, then you can go ahead and rewrite its value as one. So on the left hand side here we have log base 13 of 13. The base of the logarithms and the inside argument match each other. So we can rewrite this left hand portion as one. So the simplified version is going to be one plus the log base of 13 X. And this is going to be the completely simplified version of our given logarithms. And that means that the solution to this problem is going to be C. So I hope this video helps you in understanding how to use the property of multiplication, and I'll go ahead and see you all in the next video.

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.log7 (7x)

Key Concepts

Logarithmic Properties

Change of Base Formula

Evaluating Logarithmic Expressions

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