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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 100
Chapter 5, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
Textbook Question

Solve each equation. ln 3−ln(x+5)−ln x=0

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Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log3 (7) = 1/[log7 (3)]

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Solve each equation. 5x2−12=252x5^{x^2 - 12} = 25^{2x}

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Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log6 [4(x + 1)] = log6 (4) + log6 (x + 1)

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Solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0

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