Video: Simplifying Expressions with the Natural Log

Simplify ln 3 + 2 ln 𝑥 − ln 4.

01:49

Video Transcript

Simplify the natural log of three plus two times the natural log of 𝑥 minus the natural log of four.

In order to do this, we’re going to need to consider the properties of natural log. We know that the natural log of 𝑥 times 𝑦 equals the natural log of 𝑥 plus the natural log of 𝑦. We also know that the natural log of 𝑥 over 𝑦 is equal to the natural log of 𝑥 minus the natural log of 𝑦. We can say that the natural log of 𝑥 to the 𝑦 power equals 𝑦 times the natural log of 𝑥.

We want to take these properties and simplify what we’ve been given. Looking at this middle term, we’re given two times the natural log of 𝑥. And based on this third property, we can rewrite that as the natural log of 𝑥 squared. In the next row, we’ll just bring everything else down. And then, we’ll consider the natural log of three plus the natural log of 𝑥 squared. This is the first rule.

We simplify the natural log of three plus the natural log of 𝑥 squared by saying the natural log of three times 𝑥 squared. And we’ll again bring down the remaining bit of the problem. Natural log of three 𝑥 squared minus the natural log of four fits our second rule. When we’re subtracting a natural log from a natural log, we’re taking the natural log of the first term divided by the second term, here, three 𝑥 squared divided by four.

The simplified form of this expression is the natural log of three 𝑥 squared over four.

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