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Chapter 12: Problem 24

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{5}\left(\frac{\sqrt{x}}{25}\right)$$

Short Answer

Expert verified
\(\frac{1}{2} * \log _{5} x - 2\)

Step by step solution

01

Apply Quotient Rule

The quotient rule for logarithms states that \(\log_b \frac{M}{N} = \log_b(M) - \log_b(N)\). Thus, the given expression \(\log _{5}\left(\frac{\sqrt{x}}{25}\right)\) can be rewritten as: \(\log _{5} \sqrt{x} - \log _{5} 25\).
02

Apply Power Rule and Change of Base Formula

The power rule for logarithms states that \(\log_b M^n = n * \log_b(M)\). Similarly, the property of changing the base of the logarithm states that \(\log_b a = \log_c a / \log_c b, for any positive number c != 1.\) Thus, the above expression can be further simplified to: \(\frac{1}{2} * \log _{5} x - 2.\)
03

Express Logarithm in Expanded Form

Applying the properties of logarithms, the expanded form of the provided logarithmic expression is thus: \(\frac{1}{2} * \log _{5} x - 2.\)

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Most popular questions from this chapter

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months is modeled by the human memory function \(f(t)=75-10 \log (t+1),\) where \(0 \leq t \leq 12\) Use a graphing utility to graph the function. Then determine how many months will elapse before the average score falls below 65

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

Solve each equation. $$3^{x+2} \cdot 3^{x}=81$$

Graph: \(5 x-2 y>10\)

Describe the following property using words: \(\log _{b} b^{x}=x\)
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