SOLUTION: Rewrite the following equation in terms of base e. Round any natural logarithms to three decimal places. y=3.4(0.4)^x Type the model in terms of base e y= Almost done wi

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Rewrite the following equation in terms of base e. Round any natural logarithms to three decimal places. y=3.4(0.4)^x Type the model in terms of base e y= Almost done wi      Log On


   

Question 865078: Rewrite the following equation in terms of base e. Round any natural logarithms to three decimal places.
y=3.4(0.4)^x
Type the model in terms of base e
y=

Almost done with homework! Thank you for your help! Much appreciated!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Exponential expressions with natural base e are of the form y+=+a%2Ae%5E%28kx%29. 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 %280.4%29%5Ex equal to e%5E%28kx%29 and solve for k


%280.4%29%5Ex+=+e%5E%28kx%29


%280.4%29%5Ex+=+%28e%5Ek%29%5Ex


0.4+=+e%5Ek The two sides have the same exponent of x. So the two bases are 0.4 and e%5Ek must be equal.


ln%280.4%29+=+k Convert to logarithmic form


k+=+ln%280.4%29


k+=+-0.91629073187 Using a calculator here.


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


This means


e%5E%28kx%29=e%5E%28-0.916x%29


and we replace %280.4%29%5Ex with e%5E%28-0.916x%29 to go from


y+=+3.4%280.4%29%5Ex


to


y+=+3.4%2Ae%5E%28-0.916x%29


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Final Answer:


y+=+3.4%2Ae%5E%28-0.916x%29