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Chapter 6: Problem 5

Express \(3^{x}, x^{\pi},\) and \(x^{\sin x}\) using the base \(e\).

Short Answer

Expert verified
Question: Express the following expressions in terms of the base \(e\): \(3^x\), \(x^\pi\), and \(x^{\sin x}\). Answer: The expressions can be expressed in terms of base \(e\) as follows: - \(3^x = e^{x\ln{3}}\) - \(x^\pi = e^{\pi\ln{x}}\) - \(x^{\sin x} = e^{\sin x\ln{x}}\)

Step by step solution

01

Express \(3^x\) in terms of base \(e\)

We have the given expression: \(3^x\). Apply the property of logarithms mentioned above and get: $$3^x = e^{\ln{(3^x)}} = e^{x\ln{3}}.$$ So, \(3^x\) can be expressed as \(e^{x\ln{3}}\).
02

Express \(x^\pi\) in terms of base \(e\)

We have the given expression: \(x^\pi\). Apply the property of logarithms mentioned above and get: $$x^\pi = e^{\ln{(x^\pi)}} = e^{\pi\ln{x}}.$$ So, \(x^\pi\) can be expressed as \(e^{\pi\ln{x}}\).
03

Express \(x^{\sin x}\) in terms of base \(e\)

We have the given expression: \(x^{\sin x}\). Apply the property of logarithms mentioned above and get: $$x^{\sin x} = e^{\ln{(x^{\sin x})}} = e^{\sin x\ln{x}}.$$ So, \(x^{\sin x}\) can be expressed as \(e^{\sin x\ln{x}}\). In conclusion, the given expressions can be expressed as follows: - \(3^x = e^{x\ln{3}}\) - \(x^\pi = e^{\pi\ln{x}}\) - \(x^{\sin x} = e^{\sin x\ln{x}}\)

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Most popular questions from this chapter

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