13. For what x\u003e0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Start by writing the given equation clearly: $$x^{x^{x}$$} = $$(x^{x}$$)^{x}$$ for x$$ \u003e 0$$. Rewrite the right-hand side using the power of a power rule: (x$$^{x}$$)^{x}$$ = $$x^{x \cdot x}$$ = $$x^{x^{2}$$}$$. Now the equation becomes x$$^{x^{x}$$} = x$$^{x^{2}$$}. $$Since the bases are the same and $$x \u003e 0$$, set the exponents equal: $$x^{x}$$ = $$x^{2}. $$Rewrite the equation $$x^{x}$$ = $$x^{2} by $$taking the natural logarithm of both sides: $$\ln(x$$^{x}$$) = \ln(x$$^{2}$$). $$Use the logarithm power rule to simplify: $$x \ln(x) = 2 \ln(x). $$Since $$x \u003e 0$$, consider cases where $$\ln(x) \neq 0$$ and solve for $$x.$$

Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

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Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.AAE.13

Chapter 7, Problem 7.AAE.13

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Verified step by step guidance

Start by writing the given equation clearly: $x^{x^{x}} = (x^{x})^{x}$ for $x > 0$.

Rewrite the right-hand side using the power of a power rule: $(x^{x})^{x} = x^{x \cdot x} = x^{x^{2}}$.

Now the equation becomes $x^{x^{x}} = x^{x^{2}}$. Since the bases are the same and $x > 0$, set the exponents equal: $x^{x} = x^{2}$.

Rewrite the equation $x^{x} = x^{2}$ by taking the natural logarithm of both sides: $\ln(x^{x}) = \ln(x^{2})$.

Use the logarithm power rule to simplify: $x \ln(x) = 2 \ln(x)$. Since $x > 0$, consider cases where $\ln(x) \neq 0$ and solve for $x$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Exponents

Understanding how to manipulate and simplify expressions with exponents is crucial. Key rules include (a^b)^c = a^(bc) and a^b * a^c = a^(b+c). These properties help rewrite and compare expressions like x^(x^x) and (x^x)^x by expressing them with a common base or exponent form.

Exponentiation with Variable Exponents

Exponentiation where the exponent itself is a function of the variable, such as x^x or x^(x^x), requires careful handling. Recognizing the hierarchy of operations and how to interpret nested exponents is essential to correctly simplify and analyze the expressions.

Equation Solving and Domain Considerations

Solving the equation x^(x^x) = (x^x)^x involves setting the expressions equal and simplifying to find x > 0. Considering the domain ensures the expressions are defined, and applying logarithms or exponent rules helps isolate x and determine all valid solutions.

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Textbook Question

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

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Textbook Question

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

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Textbook Question

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

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Textbook Question

Find the limits in Exercises 1–6.

5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))

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Textbook Question

19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.

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Textbook Question

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.

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