https://youtu.be/2SCZzFy2b2s

In this video I go over some laws of logarithms and prove the law ln(xy) = ln(x) + ln(y). In my earlier videos on logs and their properties I proved this same law but using the definition of logarithms as the inverse of the exponential function. In this video, however, I show how to prove logarithm laws using integrals and derivatives like I explained in my last video.

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# Laws of Logarithms: ln(xy) = ln(x) + ln(y)

![Laws of Logarithms lnxy.jpeg](https://files.peakd.com/file/peakd-hive/mes/kh1GN46v-Laws20of20Logarithms20lnxy.jpeg)

If x and y are positive numbers and r is a rational number, then

1. ln(xy) = ln(x) + ln(y)
2. ln(x/y) = ln(x) – ln(y)
3. ln(x<sup>r</sup>) = rln(x)

**Proof:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/4EOJx5fi-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/nZxtjMtI-image.png)