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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 27
Chapter 5, Problem 27

Solve each equation. x=2log29x = 2^{\(\log\)_2 9}

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1
Recognize that the expression involves an exponent with a logarithm: \(x = 2^{\log_2 9}\). The base of the exponent and the base of the logarithm are the same (both 2).
Recall the logarithmic identity: \(a^{\log_a b} = b\). This means that when the base of the exponent and the logarithm match, the expression simplifies directly to the argument of the logarithm.
Apply the identity to simplify \(2^{\log_2 9}\) to just 9.
Therefore, the solution to the equation is \(x = 9\).
Verify the solution by substituting back into the original expression to ensure consistency.

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Key Concepts

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

Logarithmic and Exponential Functions

Logarithms and exponentials are inverse operations. The logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this inverse relationship helps simplify expressions like 2^log2(9).
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Graphs of Logarithmic Functions

Properties of Logarithms

Key properties such as log_b(b^x) = x and b^{log_b(x)} = x allow simplification of expressions involving logs and exponents with the same base. These properties are essential to solve equations like x = 2^{log2(9)}.
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Change of Base Property

Evaluating Expressions with Same Base

When the base of the exponent and the base of the logarithm are the same, the expression simplifies directly to the argument of the logarithm. For example, 2^{log2(9)} simplifies to 9, which is crucial for solving the given equation.
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Evaluating Algebraic Expressions
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