In this video I go over another law of exponent and prove that (e^x)^r = e^{r x} where r is a rational number. In this proof I use a similar method as in my previous videos on the laws of exponents and utilize the natural logarithm function, ln(x), in simplifying and breaking down the function to obtain the exponent law.

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Laws of Exponents: (e^x)^r = e^{r x}

If x and y are real numbers and r is rational, then:

1. e^x+y = e^xe^y
2. e^x-y = e^x/e^y
3. (e^x)^r = e^{r x}

Proof:

Ln is a one-to-one function: meaning that each y-value has exactly one x-value.

In fact this holds true if r is any real number (will show in later videos).