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

Problema 7.AAE.13

Capitolo 7, Problema 7.AAE.13

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

Guida verificata passo dopo passo

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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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.

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Properties of Functions

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.

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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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Solving Logarithmic Equations

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