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

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Answer 1 · 2018-01-04T16:15:53

$≅ 1.02$ $~=1.02$

$2^{3 x} = 3 e^{x}$ $2^(3x)=3e^x$

${ln 2}^{3 x} = ln (3 e^{x})$ $ln2^(3x)=ln(3e^x)$

$3 x ln 2 = ln 3 + {ln e}^{x}$ $3xln2=ln3+lne^x$

$3 x \cdot 0.69314718056 = 1.09861228867 + x$ $3x*0.69314718056=1.09861228867+x$

$2.07944154168 x - x = 1.09861228867$ $2.07944154168x-x=1.09861228867$

$1.07944154168 x = 1.09861228867$ $1.07944154168x=1.09861228867$

$x = \frac{1.09861228867}{1.07944154168}$ $x=1.09861228867/1.07944154168$

$1.01775987513$ $1.01775987513$ $≅ 1.02$ $~=1.02$

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

Answer 2 · 2018-01-04T16:17:20

$x = 1.018$ $x=1.018$

$2^{3 x} = 3 e^{x}$ $2^(3x) = 3e^x$

${ln 2}^{3 x} = ln (3 e^{x})$ $ln2^(3x) = color(red)(ln(3e^x)$

$x \cdot {ln 2}^{3} - x \cdot ln (3 e) = 0$ $x*ln2^(3) -color(red)(x*ln(3e))=0$

$x ({ln 2}^{3} - ln (3 e)) = 0$ $x(ln2^(3) -ln(3e))=0$

$x = \frac{0}{{ln 2}^{3} - ln (3 e)} = 0$ $x=0/(ln2^(3) -ln(3e))=0$

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

${ln 2}^{3 x} = ln (3 e^{x})$ $ln2^(3x) = ln(3e^x)$

${ln 2}^{3 x} = {ln e}^{x} + ln 3$ $ln2^(3x) = lne^x+ln3$

$x \cdot {ln 2}^{3} - x \cdot ln e = ln 3$ $x*ln2^(3)- x*lne=ln3$

$x ({ln 2}^{3} - ln e) = ln 3$ $x(ln2^(3)-lne)=ln3$

$x = \frac{ln 3}{{ln 2}^{3} - ln e} = 1.018$ $x=ln3/(ln2^(3)-lne)=1.018$