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Chapter 6: Problem 45

Graph \(y=\log x^{3}\) and \(y=3 \log x\) on separate sets of axes. It would seem, at first glance, that by applying the power rule for logarithms, these graphs should be the same. Are they? If not, why not? (Hint: Consider the domain in each case.)

Short Answer

Expert verified
The graphs are the same for \(x > 0\), justified by domain matching through power rules and behavioral analysis.

Step by step solution

01

Analyze the Expression \(y = \log x^3\)

The expression \(y = \log x^3\) can be rewritten using the logarithm power rule as \(y = 3\log x\). This suggests that its behavior should match \(y = 3\log x\). However, rewriting it in this way assumes that \(x^3 > 0\), meaning \(x eq 0\). Furthermore, for a real log function to be defined, \(x > 0\). Therefore, the domain of \(y = \log x^3\) is \(x > 0\).
02

Analyze the Expression \(y = 3 \log x\)

The expression \(y = 3\log x\) directly reveals that the log function is defined when \(x > 0\). Thus, the domain of \(y = 3 \log x\) is also \(x > 0\).
03

Compare the Graphs and Domains

Given both functions appear to suggest the same relationship, \(3\log x\), they indeed seem similar graphically, lying on the same curve where \(x > 0\). Both graphs coincide for positive \(x\). However, if the calculation of \(x^3\) extends to considering when it is negative (i.e., negative bases raised to an odd power, conceptual in some contexts), the domain for such consideration would only exist theoretically for real logs; by typical real-world standards, complex numbers utilize rules not accounted for here.
04

Conclusion

Despite similar algebraic manipulations and results within their shared domain \(x > 0\), the context of the power law strictly limits \(\log x^3\)'s two sides to matching the real domain of the simpler \(3 \log x\). Thus, while in real analytical terms they mirror each other over explicitly positive values, they are considered equivalent in existing practical applications.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Logarithms
When graphing logarithmic functions, it's essential to understand their curves and how they behave. For instance, if we take the function \(y = \log x^3\) and rewrite it using the power rule, it transforms into \(y = 3\log x\).

To graph these functions accurately, we follow these steps:
  • Identify the domain: Only positive values of \(x\) are considered for real logarithms.
  • Determine key points: For example, at \(x = 1\), \(\log 1 = 0\), hence both graphs will pass through \(y = 0\).
  • Analyze the behavior: As \(x\) increases, the value of \(\log x\) increases, thus scaling these values by three (in \(3\log x\)) gives a steeper graph.
The tricky part is understanding that while these transformed functions seem similar, small nuances in understanding domains can alter their true representation.
Power Rule for Logarithms
The power rule for logarithms is a handy mathematical property that simplifies expressions. It states that the logarithm of a power can be expressed as the exponent multiplied by the logarithm of the base. Formally, \(\log x^n = n \log x\).

Using this rule allows us to simplify and manipulate logarithmic expressions.
  • Simplification: \( \log x^3 = 3\log x \).
  • Computation ease: Reduces complex expressions, making calculations manageable.
  • Clarification: Clearly denotes how repeated multiplication in exponents impacts the log scale.
However, it's crucial to remember that this rule applies within the domain of definable logarithms. Both expressions \(y = \log x^3\) and \(y = 3 \log x\) rely on \(x > 0\) to remain valid. Real solutions adhere strictly to the domain restrictions.
Domain of a Function
Understanding the domain is fundamental when dealing with functions, especially with logarithms. The domain of a function defines all the possible input values (\(x\) values) for which the function is defined and yields real numbers. For logarithmic functions, this generally means the argument of the logarithm must be greater than zero.

In our equations:
  • For \(y = \log x^3\), we need \(x^3 > 0\). Thus, \(x > 0\).
  • For \(y = 3\log x\), the condition is simply \(x > 0\) since \(\log x\) needs \(x\) to be positive.
These imply that both functions share the same domain, \(x > 0\), which must be respected to ensure valid and real results. In practice, understanding the domain helps in avoiding scenarios where the function becomes undefined, especially when graphing or computing solutions.

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Most popular questions from this chapter

Salinity The salinity of the oceans changes with latitude and depth. In the tropics, the salinity increases on the surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$f(x)=31.5+1.1 \log (x+1)$$ models salinity to depths of 1000 meters at a latitude of 57.5". The variable x is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D .\) Global Physical Climatology, Academic Press.) Approximate analytically, to the nearest meter, the depth where the salinity equals 33.

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=-\sqrt{x^{2}-16}$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$-\frac{2}{3} \log _{5} 5 m^{2}+\frac{1}{2} \log _{5} 25 m^{2}$$

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$2^{-x}=\log x$$

Use any method (analytic or graphical) to solve each equation. $$\log x^{2}=(\log x)^{2}$$
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