Problem 151 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \((e^{\ln 3})^{x}\)

Step 3: 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 \(e^{\ln(3) \cdot x}\).