Question

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

Answer 1

`~=1.02`

Explanation:

`2^(3x)=3e^x`

`ln2^(3x)=ln(3e^x)`

`3xln2=ln3+lne^x`

`3x*0.69314718056=1.09861228867+x`

`2.07944154168x-x=1.09861228867`

`1.07944154168x=1.09861228867`

`x=1.09861228867/1.07944154168`

`1.01775987513` `~=1.02`

NOTES: `lnx` is "`1-1`" so you can plug it in the equation.

• `ln(ab)=lna+lnb`
• `lna^x=xlna`
• `lne=1`

Answer 2

`x=1.018`

Explanation:

`2^(3x) = 3e^x`

`ln2^(3x) = color(red)(ln(3e^x)`

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

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

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

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

`ln2^(3x) = ln(3e^x)`

`ln2^(3x) = lne^x+ln3`

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

`x(ln2^(3)-lne)=ln3`

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