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Chapter 3: Problem 69

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\log x+\log \left(x^{2}-1\right)-\log 7-\log (x+1)$$

Short Answer

Expert verified
The given logarithmic expression can be simplified and written as a single logarithm whose coefficient is \(1\) as \(\log \left(\frac{x^2-1}{7}\right)\).

Step by step solution

01

Write the equation

Start with the original expression:\(\log x+\log \left(x^{2}-1\right)-\log 7-\log (x+1)\)
02

Apply logarithm condensing rules

Using the property of logarithms: \(log_b M + log_b N = log_b (M \cdot N)\) and \(log_b M - log_b N = log_b (M \div N)\), the given equation can be rewritten as:\(\log x + \log (x^2 - 1) - \log 7 - \log (x + 1) = \log \left(\frac{x(x^2 - 1)}{7(x + 1)}\right)\)
03

Simplify the equation

Simplify the expression under the logarithm\(\log \left(\frac{x(x^2 - 1)}{7(x + 1)}\right)= \log \left(\frac{x^3-x}{7x + 7)}\right)\)
04

Further Simplification

Further simplify the fraction under the logarithm by cancelling terms: \(\log \left(\frac{x^2-1}{7}\right)\)

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Most popular questions from this chapter

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\).

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