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The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of a flood for three consecutive years?
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Ch. 8 - Sequences, Induction, and Probability
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 9, Problema 97
Retaining the Concepts. Expand:
Guida verificata passo dopo passo1
Rewrite the expression clearly: you have \( \log_7 \left( \frac{5 \sqrt{x}}{49 y^{10}} \right) \) multiplied by the fifth root of \( x \), which can be written as \( x^{\frac{1}{5}} \).
Express the square root and fifth root in exponential form: \( \sqrt{x} = x^{\frac{1}{2}} \) and \( x^{\frac{1}{5}} \) remains as is.
Rewrite the argument inside the logarithm using exponents: \( \frac{5 x^{\frac{1}{2}}}{49 y^{10}} \).
Use the logarithm property for division: \( \log_7 \left( \frac{A}{B} \right) = \log_7 A - \log_7 B \). So, split the log into \( \log_7 (5 x^{\frac{1}{2}}) - \log_7 (49 y^{10}) \).
Apply the logarithm product and power rules: \( \log_7 (5) + \log_7 (x^{\frac{1}{2}}) - \left( \log_7 (49) + \log_7 (y^{10}) \right) \). Then, use the power rule \( \log_b (a^c) = c \log_b (a) \) to bring down exponents.
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Properties of Logarithms
Logarithmic properties such as the product, quotient, and power rules allow us to expand or simplify logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N), and log_b(M^k) = k·log_b(M). These rules help break down complex expressions into sums and differences of simpler logs.
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5:36
Change of Base Property
Radicals and Exponents
Radicals like roots can be rewritten as fractional exponents, e.g., the fifth root of x is x^(1/5). Understanding this conversion is essential for applying logarithm power rules and simplifying expressions involving roots within logarithms.
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04:06Rational Exponents
Simplifying Algebraic Expressions Inside Logarithms
Before expanding, it is important to simplify the expression inside the logarithm by factoring and rewriting terms. This includes expressing constants as powers of the base if possible and separating products and quotients to apply logarithm rules effectively.
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05:07Simplifying Algebraic Expressions
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