How do you solve 2^3x = 3e^x ? Thanks

2 Answers
Jan 4, 2018

~=1.021.02

Explanation:

2^(3x)=3e^x23x=3ex

ln2^(3x)=ln(3e^x)ln23x=ln(3ex)

3xln2=ln3+lne^x3xln2=ln3+lnex

3x*0.69314718056=1.09861228867+x3x0.69314718056=1.09861228867+x

2.07944154168x-x=1.098612288672.07944154168xx=1.09861228867

1.07944154168x=1.098612288671.07944154168x=1.09861228867

x=1.09861228867/1.07944154168x=1.098612288671.07944154168

1.017759875131.01775987513 ~=1.021.02

NOTES: lnxlnx is "1-111" so you can plug it in the equation.

  • ln(ab)=lna+lnbln(ab)=lna+lnb
  • lna^x=xlnalnax=xlna
  • lne=1lne=1
Jan 4, 2018

x=1.018x=1.018

Explanation:

2^(3x) = 3e^x23x=3ex

ln2^(3x) = color(red)(ln(3e^x)ln23x=ln(3ex)

x*ln2^(3) -color(red)(x*ln(3e))=0xln23xln(3e)=0

x(ln2^(3) -ln(3e))=0x(ln23ln(3e))=0

x=0/(ln2^(3) -ln(3e))=0x=0ln23ln(3e)=0

Watch out for that step highlighted by red color. It's common mistake. correct version:

ln2^(3x) = ln(3e^x)ln23x=ln(3ex)

ln2^(3x) = lne^x+ln3ln23x=lnex+ln3

x*ln2^(3)- x*lne=ln3xln23xlne=ln3

x(ln2^(3)-lne)=ln3x(ln23lne)=ln3

x=ln3/(ln2^(3)-lne)=1.018x=ln3ln23lne=1.018