Domanda del libro di testo
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. log3 (7) = 1/[log7 (3)]
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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 101
Solve each equation.
Guida verificata passo dopo passo1
Recognize that both sides of the equation involve exponential expressions with base 5 or powers of 5. Rewrite 25 as a power of 5: since \(25 = 5^2\), rewrite the right side as \(25^{2x} = (5^2)^{2x}\).
Apply the power of a power property: \((a^m)^n = a^{mn}\). So, \((5^2)^{2x} = 5^{4x}\). Now the equation becomes \(5^{x^2 - 12} = 5^{4x}\).
Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(x^2 - 12 = 4x\).
Rewrite the equation to standard quadratic form by moving all terms to one side: \(x^2 - 4x - 12 = 0\).
Solve the quadratic equation \(x^2 - 4x - 12 = 0\) using factoring, completing the square, or the quadratic formula to find the values of \(x\).
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Properties of Exponents
Understanding how to manipulate expressions with exponents is essential. This includes knowing that a^(m*n) = (a^m)^n and that a^m * a^n = a^(m+n). These properties allow rewriting and simplifying exponential equations to make them easier to solve.
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04:06
Rational Exponents
Expressing Numbers with the Same Base
To solve exponential equations, it helps to rewrite both sides with the same base. For example, 25 can be expressed as 5^2. This allows setting the exponents equal to each other when the bases are the same, simplifying the equation to an algebraic form.
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4:47The Number e
Solving Quadratic Equations
After equating exponents, the resulting equation may be quadratic in form. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is necessary to find the values of the variable.
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06:08Solving Quadratic Equations by Factoring
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