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Chapter 4: Problem 151

a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

Short Answer

Expert verified
The simplification of \(e^{\ln 3}\) equals 3 and \(3^{x}\) in terms of base \(e\) is \(e^{\ln(3) \cdot x}\)

Step by step solution

01

Simplifying \(e^{\ln 3}\)

The natural logarithm (\(\ln x\)) is the inverse of the exponential function \(e^{x}\). They cancel out each other. So, when \(e\) raised to the power of \(\ln 3\), it will give us 3.
02

Rewriting \(3^{x}\) in terms of base \(e\)

Knowing that 3 is equivalent to \(e^{\ln 3}\) from the previous step, \(3^{x}\) can be rewritten as \((e^{\ln 3})^{x}\)
03

Further Simplification

Applying the product rule, which states that the multiplication of two terms with identical bases can be simplified to a term with the same base but with an exponent equal to the sum of the original exponents. Therefore, \((e^{\ln 3})^{x}\) can be written as \(e^{\ln(3) \cdot x}\).

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