Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 6870 by Tawakalitu. last updated on 31/Jul/16

Solve for x in     4^x  = ((192)/x)    Using lambert function

$${Solve}\:{for}\:{x}\:{in}\: \\ $$$$ \\ $$$$\mathrm{4}^{{x}} \:=\:\frac{\mathrm{192}}{{x}} \\ $$$$ \\ $$$${Using}\:{lambert}\:{function}\: \\ $$

Commented by Yozzii last updated on 31/Jul/16

a^(bx+c) =(d/(fx+g))     (a,b,c,d,f,g∈R, a≠1,d≠0,f≠0,b≠0,a>0)  ⇒(fx+g)e^((bx+c)lna) =d  (f((blna)/(blna))x+g)e^(clna) e^(bxlna) =d  ((fe^(clna−((gblna)/f)) )/(blna))(bxlna+((gblna)/f))e^(bxlna+((gblna)/f)) =d  bxlna+((gblna)/f)=W{((dblna)/f)e^(((gblna)/f)−clna) }  x=(1/(blna))[W{((dblna)/f)e^(((gblna)/f)−clna) }−((gblna)/f)]  e=Euler′s constant  In your problem, b=f=1,c=g=0,d=192,a=4  ⇒x=(1/(ln4))W{192ln4}

$${a}^{{bx}+{c}} =\frac{{d}}{{fx}+{g}}\:\:\:\:\:\left({a},{b},{c},{d},{f},{g}\in\mathbb{R},\:{a}\neq\mathrm{1},{d}\neq\mathrm{0},{f}\neq\mathrm{0},{b}\neq\mathrm{0},{a}>\mathrm{0}\right) \\ $$$$\Rightarrow\left({fx}+{g}\right){e}^{\left({bx}+{c}\right){lna}} ={d} \\ $$$$\left({f}\frac{{blna}}{{blna}}{x}+{g}\right){e}^{{clna}} {e}^{{bxlna}} ={d} \\ $$$$\frac{{fe}^{{clna}−\frac{{gblna}}{{f}}} }{{blna}}\left({bxlna}+\frac{{gblna}}{{f}}\right){e}^{{bxlna}+\frac{{gblna}}{{f}}} ={d} \\ $$$${bxlna}+\frac{{gblna}}{{f}}={W}\left\{\frac{{dblna}}{{f}}{e}^{\frac{{gblna}}{{f}}−{clna}} \right\} \\ $$$${x}=\frac{\mathrm{1}}{{blna}}\left[{W}\left\{\frac{{dblna}}{{f}}{e}^{\frac{{gblna}}{{f}}−{clna}} \right\}−\frac{{gblna}}{{f}}\right] \\ $$$${e}={Euler}'{s}\:{constant} \\ $$$${In}\:{your}\:{problem},\:{b}={f}=\mathrm{1},{c}={g}=\mathrm{0},{d}=\mathrm{192},{a}=\mathrm{4} \\ $$$$\Rightarrow{x}=\frac{\mathrm{1}}{{ln}\mathrm{4}}{W}\left\{\mathrm{192}{ln}\mathrm{4}\right\} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com