Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


d. 2ˣ = 2² ˡⁿ ˣ

Verified step by step guidance

1. Recall the properties of exponents and logarithms. The expression 2^{x} means 2 raised to the power x.
2. The expression 2^{2 \(\ln\) x} can be rewritten using the property a^{b} = e^{b \(\ln\) a}, so 2^{2 \(\ln\) x} = e^{(2 \(\ln\) x)(\(\ln\) 2)}.
3. Compare 2^{x} and 2^{2 \(\ln\) x} by expressing both in terms of the exponential function with base e: 2^{x} = e^{x \(\ln\) 2} and 2^{2 \(\ln\) x} = e^{2 \(\ln\) x \(\cdot\) \(\ln\) 2}.
4. Since x > 0 and y > 0, analyze whether the exponents x \(\ln\) 2 and 2 \(\ln\) x \(\cdot\) \(\ln\) 2 are equal for all x > 0. This requires checking if x = 2 \(\ln\) x holds for all x > 0.
5. Test the equality by substituting specific positive values of x to see if 2^{x} equals 2^{2 \(\ln\) x}. If they are not equal for all x, then the statement is false and the substitution serves as a counterexample.

Key Concepts

Properties of Exponents

Exponents follow specific rules, such as a^(m+n) = a^m * a^n and (a^m)^n = a^(mn). Understanding how to manipulate expressions with exponents is essential to verify or refute equations involving exponential terms.

Natural Logarithm and Its Relationship to Exponents

The natural logarithm (ln) is the inverse function of the exponential function with base e. It allows rewriting expressions like a^x as e^(x ln a), which is crucial for comparing or transforming exponential expressions with different bases.

Evaluating and Simplifying Exponential Expressions

To determine the truth of an equation involving exponentials, one must simplify both sides using logarithmic and exponential identities. This process often involves substituting and comparing expressions to check equality or find counterexamples.