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Write each equation in its equivalent logarithmic form. 2-4 = 1/16
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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 13
Write each equation in its equivalent logarithmic form. ∛8 = 2
Guida verificata passo dopo passo1
Identify the given exponential equation: \(\sqrt[3]{8} = 2\). This can be rewritten as \(8^{\frac{1}{3}} = 2\) because the cube root of 8 is the same as raising 8 to the power of \(\frac{1}{3}\).
Recall the relationship between exponential and logarithmic forms: if \(a^x = b\), then the equivalent logarithmic form is \(\log_{a} b = x\).
In the equation \(8^{\frac{1}{3}} = 2\), identify the base \(a = 8\), the exponent \(x = \frac{1}{3}\), and the result \(b = 2\).
Apply the logarithmic form using the identified values: write \(\log_{8} 2 = \frac{1}{3}\).
This expresses the original equation in its equivalent logarithmic form, completing the conversion.
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Exponential and Logarithmic Forms
Exponential and logarithmic forms are two ways to express the same relationship. An equation like a^b = c can be rewritten as log_a(c) = b, where the logarithm answers the question: to what power must the base a be raised to get c?
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5:02
Solving Logarithmic Equations
Cube Roots and Rational Exponents
A cube root, such as ∛8, can be expressed as an exponent of 1/3, so ∛8 = 8^(1/3). Understanding this helps convert root expressions into exponential form, which is essential for rewriting equations in logarithmic form.
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04:06Rational Exponents
Properties of Logarithms
Logarithms have properties that allow simplification and conversion between forms. Recognizing that log_b(c) = x means b^x = c is key to rewriting equations, especially when dealing with roots and powers.
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5:36Change of Base Property
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