Question 865078
Exponential expressions with natural base e are of the form {{{y = a*e^(kx)}}}. The value of 'a' is already set to be 3.4 since this is the coefficient of the given equation. We just need to find k



So set {{{(0.4)^x}}} equal to {{{e^(kx)}}} and solve for k



{{{(0.4)^x = e^(kx)}}}



{{{(0.4)^x = (e^k)^x}}}



{{{0.4 = e^k}}} The two sides have the same exponent of x. So the two bases are 0.4 and {{{e^k}}} must be equal.



{{{ln(0.4) = k}}} Convert to logarithmic form



{{{k = ln(0.4)}}}



{{{k = -0.91629073187}}} Using a <a href="https://www.google.com/search?q=ln(0.4)">calculator</a> here.



{{{k = -0.916}}} Round any natural logarithms to three decimal places. 



This means



{{{e^(kx)=e^(-0.916x)}}}



and we replace {{{(0.4)^x}}} with {{{e^(-0.916x)}}} to go from



{{{y = 3.4(0.4)^x}}}



to



{{{y = 3.4*e^(-0.916x)}}}



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Final Answer:



{{{y = 3.4*e^(-0.916x)}}}