Question

What does multiplication translate to when using logs?

Answer 1

It translates to addition.

Answer 2

General Idea:

We can use the product rule for exponentiation to derive a corresponding product rule for logarithms.

`e^a \cdot e^b=e^(a+b)`

Also we know
`e^(ln(m))=m`

Using the above rule we can write the below equation...

`e^(ln(xy))=xy \; \rightarrow 1^(st) \; equation\\ \\ e^(ln(x))=x \; \rightarrow \; 2^(nd) \; equation\\ \\ e^(ln(y))=y \; \rightarrow \; 3^(rd) \; equation\\ \\ Substituting \; 2^(st) \; equation \; and \; 3^(rd) \; equation\; in\; 1^(st)\; equation, \; we\; get...\\ \\ e^(ln(xy))=e^(ln(x)) \cdot e^(ln(y))\\ \\ Applying \; the \; formula\; of\; exponent\; in\; right\; sides\; of\; equation\\ \\ e^(ln(xy))=e^(ln(x)+ln(y))\\ \\ Taking \; natural \; logarithm\; on\; both\; sides\\ \\`

`ln[e^(ln(xy))]=ln[e^(ln(x)+ln(y))]\\ \\ And \; using\; the\; formula\; lnm^n=n(lnm), we\; get...\\ \\ ln(xy)=ln(x)+ln(y)`

Conclusion:

The product rule for exponentiation

`e^(x) \cdot e^(y)=e^(x+y)`

The product rule of logarithms

`ln(xy)=ln(x)+ln(y)`