Domanda del libro di testo
Exercises 137–139 will help you prepare for the material covered in the next section. Solve for x: a(x - 2) = b(2x + 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 128
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
logb (xy)5 = (logb x + logb y)5
Guida verificata passo dopo passo1
Recall the logarithm property for the logarithm of a product: . This means the log of a product is the sum of the logs.
Apply the power rule of logarithms: . This means the exponent can be brought in front as a multiplier.
Rewrite the left side of the equation using these properties: .
Next, use the product rule inside the logarithm: .
Compare this with the right side of the original statement: . Notice that the original statement raises the sum to the fifth power, which is different from multiplying the sum by 5. Therefore, the original statement is false, and the correct statement is .
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Properties of Logarithms
Logarithms have specific properties that simplify expressions, such as the product rule: log_b(xy) = log_b(x) + log_b(y), and the power rule: log_b(x^n) = n * log_b(x). Understanding these rules is essential to manipulate and evaluate logarithmic expressions correctly.
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5:36
Change of Base Property
Exponentiation and Logarithms
When an exponent is applied to a product inside a logarithm, the power rule allows the exponent to be factored out: log_b((xy)^5) = 5 * log_b(xy). This differs from raising the sum of logarithms to a power, which is not equivalent, highlighting the importance of correctly applying exponent rules in logarithmic contexts.
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5:02Solving Logarithmic Equations
Distinguishing Between Expressions and Their Equivalents
It is crucial to recognize that (log_b x + log_b y)^5 is not the same as 5 * (log_b x + log_b y). The former raises the entire sum to the fifth power, while the latter multiplies the sum by 5. This distinction helps avoid common mistakes when simplifying or verifying logarithmic statements.
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05:09Introduction to Algebraic Expressions
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In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]
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In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
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Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.
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If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.
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In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
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