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Graph f(x) = 2x and its inverse function in the same rectangular coordinate system.
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 5, Problema 93
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.
x log 10x = x2
Guida verificata passo dopo passo1
Recall the logarithm property: because the base of the logarithm is 10 and is the argument.
Rewrite the left side of the equation using this property: .
Compare the left side and the right side of the original equation: both simplify to , so the equation is true for all real values of .
Since the equation is true as given, no changes are necessary to make it true.
To summarize, the key step is recognizing the logarithm property that , which simplifies the expression and confirms the equality.
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Properties of Logarithms
Logarithms have specific properties that simplify expressions, such as log_b(b^x) = x. Understanding these properties allows you to rewrite and evaluate logarithmic expressions accurately, which is essential for verifying equations involving logs.
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Change of Base Property
Evaluating Logarithmic Expressions
Evaluating logarithmic expressions involves applying the definition of logarithms and their properties to simplify or compute values. For example, recognizing that log 10^x equals x when the base is 10 helps in simplifying the given equation.
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Equation Verification and Manipulation
Verifying an equation requires substituting expressions and simplifying both sides to check equality. If false, algebraic manipulation is used to adjust terms and produce a true statement, ensuring a clear understanding of the relationship between variables.
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06:00Categorizing Linear Equations
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