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Evaluate or simplify each expression without using a calculator. In e6
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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 87
Let logb 2 = A and logb 3 = C and Write each expression in terms of A and C. logb √(2/27)
Guida verificata passo dopo passo1
Start by expressing the argument inside the logarithm in terms of powers: \( \sqrt{\frac{2}{27}} = \left( \frac{2}{27} \right)^{\frac{1}{2}} \).
Rewrite the fraction inside the parentheses as a quotient of powers: \( \left( \frac{2}{3^3} \right)^{\frac{1}{2}} \) since \(27 = 3^3\).
Apply the logarithm power rule: \( \log_b \left( \frac{2}{3^3} \right)^{\frac{1}{2}} = \frac{1}{2} \log_b \left( \frac{2}{3^3} \right) \).
Use the logarithm quotient rule: \( \log_b \left( \frac{2}{3^3} \right) = \log_b 2 - \log_b 3^3 \).
Apply the logarithm power rule again: \( \log_b 3^3 = 3 \log_b 3 \). Now substitute \( \log_b 2 = A \) and \( \log_b 3 = C \) to write the expression entirely in terms of \( A \) and \( C \).
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Properties of Logarithms
Logarithms have specific properties such as the product, quotient, and power rules. These allow us to rewrite complex logarithmic expressions as sums, differences, or multiples of simpler logarithms, which is essential for expressing the given log in terms of A and C.
Video consigliato:
5:36
Change of Base Property
Change of Base and Given Logarithm Values
The problem provides logb 2 = A and logb 3 = C, which means logarithms of 2 and 3 with base b are known. Using these values, we can express other logarithms involving 2 and 3 by breaking down numbers into prime factors and substituting A and C accordingly.
Video consigliato:
5:36Change of Base Property
Simplifying Radicals and Fractions Inside Logarithms
The expression involves a square root and a fraction inside the logarithm. Understanding how to rewrite radicals as fractional exponents and how to separate logarithms of quotients helps in breaking down the expression into parts that can be expressed using A and C.
Video consigliato:
05:45Radical Expressions with Fractions
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Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−4)+ln(x+1)=ln(x−8)
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Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log x+log(x+3)=log 10
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