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Chapter 3: Problem 6

Explain why \(b^{x}=e^{x \ln b}\).

Short Answer

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Question: Show that \(b^x = e^{x \ln b}\). Answer: To demonstrate that \(b^x = e^{x \ln b}\), we first recalled the definition of logarithm and exponential function. We used the properties of logarithms to rewrite the expression on the right-hand side of the equation, and showed that this rewritten expression is equivalent to the one on the left. Therefore, we proved that \(b^x = e^{x \ln b}\).

Step by step solution

01

Recall Definitions of Logarithms and Exponential Functions

The logarithm function with base \(b\) is denoted as \(\log_b(x)\) and is the inverse of the exponential function with base \(b\), denoted as \(b^x\). In other words, if \(y = \log_b(x)\), then \(x = b^y\). Meanwhile, the natural logarithm function is a special case of the logarithm with base \(e\), which is approximately 2.718, and is denoted as \(\ln(x)\).
02

Use Logarithm Properties to Rewrite the Expression on the Right

We start with the expression on the right side of the equation: \(e^{x \ln b}\). Next, we recall the logarithm property: \(a^{\log_a(x)} = x\). Using this property, we can rewrite the expression on the right side as \(b^{x\log_b(e)}\). Since \(x\) and \(\log_b(e)\) are multiplied together, we use the property of logarithms \(y\log_a(x)=\log_a(x^y)\) on the exponents: $$e^{x \ln b} = b^{x\log_b(e)} = b^{\log_b(e^x)}$$
03

Show the Rewritten Expression is Equivalent to the One on the Left

Finally, we recall the definition of the logarithm function, particularly its inverse relationship with the exponential function. Using the property \(a^{\log_a(x)}=x\), we can simplify the expression we obtained in Step 2: $$b^{\log_b(e^x)} = e^x$$ Since the expression \(b^x\) on the left side of the equation is equivalent to the right side, we have proven that \(b^x = e^{x \ln b}\).

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