Domanda del libro di testo
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
Solve each equation. 52x ⋅ 54x=125
Guida verificata passo dopo passo1
Recognize that the equation involves exponential expressions with the same base. The equation is \(5^{2x} \cdot 5^{4x} = 125\).
Use the property of exponents that states when multiplying like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). So rewrite the left side as \(5^{2x + 4x} = 5^{6x}\).
Rewrite the right side, 125, as a power of 5. Since \(125 = 5^3\), the equation becomes \(5^{6x} = 5^3\).
Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(6x = 3\).
Solve the equation \(6x = 3\) for \(x\) by dividing both sides by 6: \(x = \frac{3}{6}\).
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Properties of Exponents
This concept involves rules for manipulating expressions with exponents, such as multiplying powers with the same base by adding their exponents. For example, 5^(2x) * 5^(4x) equals 5^(2x + 4x) = 5^(6x). Understanding these properties simplifies solving exponential equations.
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Rational Exponents
Expressing Numbers with the Same Base
To solve exponential equations, it helps to rewrite all terms with the same base. Since 125 can be written as 5^3, the equation 5^(6x) = 125 becomes 5^(6x) = 5^3, allowing us to set the exponents equal to each other for solving.
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Solving Linear Equations
After equating the exponents, the problem reduces to solving a linear equation like 6x = 3. This involves isolating the variable by performing inverse operations, such as dividing both sides by 6, to find the value of x.
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04:02Solving Linear Equations with Fractions
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