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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 99
Capitolo 5, Problema 99

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

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Recognize that the expression is \(10^{(\log \sqrt{x})}\), where the base of the logarithm is 10 (common logarithm).
Recall the property of logarithms and exponents: \(a^{\log_a b} = b\). Here, the base of the exponent and the logarithm match (both base 10), so this property applies.
Rewrite the expression using the property: \(10^{(\log \sqrt{x})} = \sqrt{x}\).
Express the square root in exponential form: \(\sqrt{x} = x^{\frac{1}{2}}\).
Thus, the simplified form of the expression is \(x^{\frac{1}{2}}\).

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Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as log(a^b) = b·log(a). Understanding how to manipulate logarithmic expressions, including the use of roots and exponents, is essential for simplifying expressions like log(√x).
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Change of Base Property

Relationship Between Exponents and Logarithms

Exponents and logarithms are inverse operations. For example, 10^(log y) = y when the base of the logarithm matches the base of the exponent. This property allows simplification of expressions like 10^(log √x) directly to √x.
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Rational Exponents

Simplifying Radicals

A radical such as √x can be expressed as an exponent, x^(1/2). Recognizing this helps in rewriting expressions involving roots into exponential form, which can then be combined with logarithmic properties for simplification.
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Adding & Subtracting Unlike Radicals by Simplifying
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