Evaluate or simplify each expression without using a calculator. 10^{log ∛x}

검증된 단계별 안내

Recognize that the expression is \(10^{(\log \sqrt[3]{x})}\), where \(\log\) denotes the logarithm base 10.

Recall the property of logarithms and exponents: \(a^{\log_a b} = b\). Here, the base of the exponent and the base of the logarithm are both 10, so this property applies.

Rewrite the expression inside the logarithm: \(\sqrt[3]{x} = x^{\frac{1}{3}}\).

Apply the logarithm power rule: \(\log(x^{\frac{1}{3}}) = \frac{1}{3} \log x\).

Use the exponent and logarithm property to simplify: \(10^{\log \sqrt[3]{x}} = 10^{\frac{1}{3} \log x} = (10^{\log x})^{\frac{1}{3}} = x^{\frac{1}{3}}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the power rule: log(a^b) = b·log(a). Understanding these properties allows you to rewrite and simplify complex logarithmic expressions effectively.

Relationship Between Exponents and Logarithms

Exponents and logarithms are inverse operations. For example, 10^(log x) = x when the log base is 10. This inverse relationship helps simplify expressions where an exponent is a logarithm.

Simplifying Radicals and Fractional Exponents

Radicals like ∛x can be expressed as fractional exponents (x^(1/3)). Converting radicals to fractional exponents makes it easier to apply logarithmic and exponential rules during simplification.

Evaluate or simplify each expression without using a calculator. 10log ∛x