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

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Hass 15th Edition

Ch. 7 - Transcendental Functions

문제 7.AAE.13

7장, 문제 7.AAE.13

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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.