SOLUTION: How does one find the domain of these monsters? {{{H(t)}}} = {{{ ln(root(3, ( e^(-2x) (6x-10) ) / ((x^4+8)^7) )) }}} & {{{L(t)}}} = {{{ ln(sqrt( ((x^2+5)^10)/((e^(x^3

Algebra ->  Graphs -> SOLUTION: How does one find the domain of these monsters? {{{H(t)}}} = {{{ ln(root(3, ( e^(-2x) (6x-10) ) / ((x^4+8)^7) )) }}} & {{{L(t)}}} = {{{ ln(sqrt( ((x^2+5)^10)/((e^(x^3      Log On


   

Question 1006425: How does one find the domain of these monsters?

H%28t%29 = +ln%28root%283%2C+%28+e%5E%28-2x%29+%286x-10%29+%29+%2F+%28%28x%5E4%2B8%29%5E7%29+%29%29+

&
L%28t%29 = +ln%28sqrt%28+%28%28x%5E2%2B5%29%5E10%29%2F%28%28e%5E%28x%5E3%29%29%284-3x%29%29+%29%29+

Please explain
Thank you
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let us consider H(t).

The domain of this monster is defined by the condition that the expression under the logarithm is positive.

The denominator under the cubic root is always positive.  So,  the ratio under the cubic root is positive if and only if
the numerator is positive.

It,  in turn,  is determined by the term  (6x - 10),  because the factor  e%5E%28-3x%29  is always positive.

Thus the domain of  H(t) is   6x - 10 >= 0,   or   x >= 10%2F6 = 5%2F3.


Similar analysis works for  L(t).

Its domain is   4-3x > 0,   or   x < 4%2F3.