One of 'e' Logarithm Properties

The most common logarithm proofs that you can find in most books,internet,wiki,and teachers might have taught you is this.....
Prove $$ \ln (uv) = ln(u) + ln(v) $$

$$Proof:$$
Let $$ln(u)=x \; and \; ln(v)=y$$
Then $$u = e^{x} \; and \; v = e^{y}$$
$$u \* v = e^{x} \* e^{y} = e^{x+y}$$
Therefore, $$e^{ln(u) + ln(v)} = u \*v$$
$$ln(u) + ln(v) = ln(uv)$$. This completes the proof

I have another way of proofing this with calculus ...

$$Proof:$$
Let the variable $$ w=ut $$ in $$ ln(v)=\int^v_1 \frac{1}{t} dt $$
Then $$ dw=u \; dt $$ and the limits of integration $$ t=1 $$ and $$ t=v \Leftrightarrow w=u $$ and $$ w=uv $$
Hence,
$$ ln(v) = \int^{uv}_u \frac{1}{\frac{w}{u}}\frac{1}{u} \; dw = \int^{uv}_u \frac{1}{w} \; dw = \int^{uv}_u \frac{1}{t} \; dt $$
$$ln(u) + ln(v) = \int^u_1 \frac{1}{t} \; dt + \int^{uv}_u \frac{1}{t} \; dt = \int^{uv}_1 \frac{1}{t} \; \\ dt=ln(uv)$$.
The property is proved in different way! Happy proofing!