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Chapter 3: Problem 57

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=100(4.6)^{x}$$

Short Answer

Expert verified
The equation \(y = 100*(4.6)^x\) rewritten in terms of base \(e\) is \(y = 100e^{x*1.526}\).

Step by step solution

01

Change of Base for Exponentials

Firstly, apply the formula for changing the base of an exponential function. Start with the equation \(y = 100 * 4.6^x\) and rewrite it as \(y = 100 * e^{x*\ln{4.6}}\), according to the formula \(a^b = e^{b*\ln{a}}\). This change happens because \(e\) and \(\ln\) are inverses of each other.
02

Simplification of Equation

Next, simplification of the equation should take place. It can be written as \(y = 100e^{x*\ln{4.6}}\).
03

Rounding to Three Decimal Places

Finally, compute the value of \(x*\ln{4.6}\) and round it to three decimal places. If the given value of \(x\) is 1 for example, \(\ln{4.6} = 1.526056\) rounding three decimal places yields 1.526.

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