Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 191795 by 073 last updated on 30/Apr/23

3^x −2^x =19  x=??

3x2x=19x=??

Commented by mr W last updated on 30/Apr/23

we know 27−8=19,  i.e. 3^3 −2^3 =19  therefore x=3.  there is no other smart method you  can blindly apply to calculate the  root. for example you can′t solve  3^x −2^x =20 exactly.

weknow278=19,i.e.3323=19thereforex=3.thereisnoothersmartmethodyoucanblindlyapplytocalculatetheroot.forexampleyoucantsolve3x2x=20exactly.

Answered by deleteduser1 last updated on 30/Apr/23

x=3  It remains to show that no other solution exists  3^x =2^x +19; obviously, if there exists another x  Then x>3  Observe that (d/dx)(3^x )=3^x In(3)  and (d/dx)(2^x +19)=2^x In(2)  And since 3^x In(3)>2^x In(2) for x>3  3^x >2^x +19 for x>3. The two curves never   intersect again. Hence,x=3 is the only solution.

x=3Itremainstoshowthatnoothersolutionexists3x=2x+19;obviously,ifthereexistsanotherxThenx>3Observethatddx(3x)=3xIn(3)andddx(2x+19)=2xIn(2)Andsince3xIn(3)>2xIn(2)forx>33x>2x+19forx>3.Thetwocurvesneverintersectagain.Hence,x=3istheonlysolution.

Terms of Service

Privacy Policy

Contact: info@tinkutara.com