Question 1138686


a. 

{{{6^(2x)=250}}}....take a log of both sides

{{{log(6^(2x))=log(250)}}}

{{{2x*log(6)=log(2*5^3)}}}............since {{{log(6)=log(2*3)=log(2)+log(3)}}}, and {{{log(2*5^3)=log(2) + 3log(5)}}}, we have

{{{2x(log(2)+log(3))=log(2) + 3log(5)}}}

{{{x=(log(2) + 3log(5))/2(log(2)+log(3))}}}

{{{x=(0.69314718 + 4.8283137373)/2(0.69314718+1.098612288668)}}}

{{{x=5.5214609173/3.583518937336}}}

{{{x=1.5408}}}


b.

{{{log(3, (x-3))+log(3 ,(x-7))=4}}} .....convert to base {{{10}}}

{{{log((x-3))/log(3)+log((x-7))/log(3)=4}}} ....both sides multiply by {{{log(3)}}}

{{{log((x-3))+log((x-7))=4 log(3)}}}

{{{log((x-3)(x-7))= log(3^4)}}}.......since log same, we have

{{{(x-3)(x-7)= 3^4}}}

{{{x^2-7x-3x+21= 81}}}

{{{x^2-10x+21- 81=0}}}

{{{x^2-10x- 60=0}}}......use quadratic formula

{{{x=(-b+-sqrt(b^2-4ac))/2a}}}

{{{x=(-(-10)+-sqrt((-10)^2-4*1*(-60)))/(2*1)}}}

{{{x=(10+-sqrt(100+240))/2}}}

{{{x=(10+-sqrt(340))/2}}}

{{{x=(10+-18.4391)/2}}}

{{{x=(5+-9.2195)}}}

solutions:

{{{x=5+9.2195}}}=>{{{x=14.2195}}}
{{{x=5-9.2195}}}=>{{{x= -4.2195}}}=> since log disregard negative solution

so, your solution is:{{{x=14.2195}}}



Given the function  {{{f(x)= log(2, (x-2))-1 }}}

1. Graph the function

{{{ graph( 600, 600, -10, 10, -10, 10, log(2, (x-2))-1) }}}

2. Find its domain & range

domain: { {{{x}}} element {{{R}}} : {{{x>2}}} }

range: {{{R}}} (all real numbers)

3. Write the equation of its asymptote

Vertical asymptote: {{{x=2}}}

{{{log(x - 2)/log(2) - 1}}}->{{{-infinity}}} as {{{x->2}}}

{{{drawing ( 600, 600, -10, 10, -10, 10,
line(2,-10,2,10),
graph( 600, 600, -10, 10, -10, 10, log(2, (x-2))-1)) }}}


Horizontal asymptote: {{{none }}}