Question 53295
I answered this one using natural logs earlier today.  I hope you're not the same person and I just wasn't clear enough.  I will explain myself:
{{{4^(x-1)*3^(x+2)=12^4}}}
Take the natural log of both sides of the equation:
{{{ln(4^(x-1)*3^(x+2))=ln(12^4)}}}
{{{ln(a*b)=ln(a)+ln(b)}}}so...
{{{ln(4^(x-1))+ln(3^(x+2))=ln(12^4)}}}
{{{ln(a^b)=bln(a)}}} so...
{{{(x-1)ln(4)+(x+2)ln(3)=ln(12^4)}}}
Distribute ln(4) and ln(3).
{{{xln(4)-ln(4)+xln(3)+2ln(3)=ln(12^4)}}}
Add ln(4) and subtract 2ln(3) from both sides of the equation:
{{{xln(4)-ln(4)+ln(4)+xln(3)+2ln(3)-2ln(3)=ln(12^4)+ln(4)-2ln(3)}}}
{{{xln(4)+xln(3)=ln(12^4)+ln(4)-2ln(3)}}}
Factor out x on the left side of the equation and use the fact that blna=lna^b to simplify the right side:
{{{x(ln(4)+ln(3))=ln(12^4)+ln(4)-ln(3^2)}}}
Use the fact that lna+lnb=ln(a*b) and lna-lnb=ln(a/b)to simplify the equation:
{{{x(ln(4*3))=ln((12^4*4)/(3^2))}}}
{{{x(ln(12))=ln(20736*4/9)}}}
{{{x(ln(12))=ln(82944/9)}}}
{{{xln(12)=ln(9216)}}}
Divide both sides by ln(12)
{{{(xln(12))/(ln(12))=(ln(9216))/(ln(12))}}}
{{{x=(ln(9216))/(ln(12))}}}
Stick ln(9216) in your calculator=9.128696383.
Stick ln(12) in your calculator=2.48490665.
so
{{{x=9.128696383/2.48490665}}}
{{{x=3.673657675}}}
You can verify this with a graphing calculator by finding the intersection between 4^(x-1)*3(x+2) and 12^4.
If you plug your answer into the equation you get:
20736.00006=20736 which is very close.  Keep in mind that we had to round off.