Determine whether the following statements are true and give an explanation or counterexample.

If y= 3ˣ , then x = ³√y

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Consider the function \( y = 3^x \).

To solve for \( x \) in terms of \( y \), take the logarithm of both sides: \( \log(y) = \log(3^x) \).

Apply the logarithmic identity \( \log(a^b) = b \log(a) \) to get \( \log(y) = x \log(3) \).

Solve for \( x \) by dividing both sides by \( \log(3) \): \( x = \frac{\log(y)}{\log(3)} \).

Compare this expression with \( x = \sqrt[3]{y} \) to determine if they are equivalent.

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주요 개념

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

Exponential Functions

Exponential functions are mathematical expressions in the form y = a^x, where 'a' is a positive constant and 'x' is the variable exponent. In this case, y = 3^x represents an exponential function where the base is 3. Understanding the properties of exponential functions is crucial for manipulating and solving equations involving them.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For an exponential function like y = 3^x, the inverse is found by solving for x in terms of y, leading to x = log₃(y). This concept is vital for determining relationships between variables and understanding how to express one variable in terms of another.

Cube Root

The cube root of a number y, denoted as ³√y, is a value that, when multiplied by itself three times, gives y. This concept is important when analyzing the statement x = ³√y, as it implies a specific relationship between x and y that must be verified against the original exponential equation.

Determine whether the following statements are true and give an explanation or counterexample.If y= 3ˣ , then x = ³√y