Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 93

Chapter 5, Problem 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 10^x = x²

Verified step by step guidance

Recall the logarithm property:

\log (10^{x}) = x

because the base of the logarithm is 10 and

10^x

is the argument.

Rewrite the left side of the equation using this property:

x \, \log (10^{x}) = x \, x = x^{2}

Compare the left side and the right side of the original equation: both simplify to

x^{2}

, so the equation is true for all real values of

x

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

\log (10^{x}) = x

, which simplifies the expression and confirms the equality.

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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_b(b^x) = x. Understanding these properties allows you to rewrite and evaluate logarithmic expressions accurately, which is essential for verifying equations involving logs.

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.

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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Related Practice

Textbook Question

Graph f(x) = 2^x and its inverse function in the same rectangular coordinate system.

Textbook Question

n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 73(2.6)^x

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Textbook Question

Solve each equation. 3^x+2 ⋅ 3^x=81

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Textbook Question

Evaluate or simplify each expression without using a calculator. e^{ln 125}

Textbook Question

Solve each equation. 5^2x ⋅ 5^4x=125

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Textbook Question

n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 6.5(0.43)^x

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

Verified video answer for a similar problem:

Key Concepts

Properties of Logarithms

Evaluating Logarithmic Expressions

Equation Verification and Manipulation

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 10^x = x²