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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 100
Capitolo 5, Problema 100

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

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1
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: \(10^{\log \sqrt[3]{x}} = 10^{\frac{1}{3} \log x} = (10^{\log x})^{\frac{1}{3}} = x^{\frac{1}{3}}\).

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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.
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Change of Base Property

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.
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Rational Exponents

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.
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Radical Expressions with Fractions
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