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Chapter 4: Problem 123

Describe the following property using words: \(\log _{b} b^{x}=x\).

Short Answer

Expert verified
The property \(\log _{b} b^{x}=x\) is a fundamental property of logarithms. It expresses that the logarithm base \(b\) of \(b\) to the power \(x\) returns \(x\). This essentially tells us that the exponent \(x\) that we use to raise our base \(b\) to will always be returned when we take the logarithm (with the same base) of the result.

Step by step solution

01

Understanding the Logarithmic Function

To begin with, let's first understand what a logarithmic function is. The logarithmic function \( \log_b{a} \) is defined as the inverse function of the exponential function, denoted \( b^y = a \). Therefore, \( y = \log_b{a} \).
02

Translating the Property

Now, translate the property \(\log _{b} b^{x}=x\). Looking at it, this can be explained as 'the logarithm base \(b\) of \(b\) to the power \(x\) equals \(x\)'. This means that \(x\) is the exponent to which you must raise the base \(b\) to obtain \(b\) to the power \(x\). This is straightforward because raising a number to an exponent and then taking the logarithm (with the matching base) will simply return the original exponent \(x\).

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Most popular questions from this chapter

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2007 to \(1,000,000\) people in \(2010 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 8000 dollar Annual Interest Rate 20.3% Accumulated Amount 12,000 dollar Time \(t\) in Years _______

Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.
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