Textbook Question
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.
Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 99
Evaluate or simplify each expression without using a calculator. 10log √x
Verified step by step guidance1
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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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 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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04:06Rational 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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5:48Adding & Subtracting Unlike Radicals by Simplifying
Related Practice
Textbook Question
Evaluate or simplify each expression without using a calculator.
Textbook Question
Evaluate or simplify each expression without using a calculator. 10log ∛x
Textbook Question
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)
Textbook Question
Solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0
Textbook Question
In Exercises 89–102, 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.