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Chapter 10: Problem 18

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single number if possible. Assume that all variables represent positive real numbers. $$ \log _{4} \frac{\sqrt[4]{z} \cdot \sqrt[5]{w}}{s^{2}} $$

Short Answer

Expert verified
\frac{\text{log}_4 (\root{4}{z}) + \text{log}_4 (\root{5}{w}) - \text{log}_4 (s^2)} becomes \(\frac{\frac{\text{log}_4 (z)}{4} + \frac{\text{log}_4 (w)}{5} - 2 \text{log}_4 (s)}{5}\)

Step by step solution

01

Apply log of division property

Using the property of logarithms that states \(\frac{a}{b} = \frac{a}{b}\)we can split the logarithm into two separate parts. \(\frac{\text{log}_4 (\frac{\root{4}{z} * \root{5}{w}}{s^2})}{\text{log}_4} = \frac{log_4 (\root{4}{z} * \root{5}{w}) - \text{log}_4 (s^2)}\)
02

Apply the log of product property

Use the property that states \(\text{log}_b (xy) = \text{log}_b (x) + \text{log}_b (y)\). This lets us separate the product inside the numerator's logarithm: \(\frac{\text{log}_4 (\root{4}{z}) + \text{log}_4 (\root{5}{w})}{- log_4 (s^2)}\)
03

Simplify the roots

Apply the property \(\text{log}_b (x^{1/n}) = \frac{\text{log}_b (x)}{n}\) to simplify the logarithms of the roots: \(\frac{\text{log}_4 (z^{1/4}) + \text{log}_4 (w^{1/5}) - log_4 (s^2)}{n}\) becomes \(\frac{\frac{\text{log}_4 (z)}{4} + \frac{\text{log}_4 (w)}{5} - 2 \text{log}_4 (s)}{n}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Rules
Logarithms have several key properties and rules that simplify complex expressions. These rules help transform a cumbersome logarithmic statement into a more manageable form.
  • Product Rule: \(\text{log}_b (xy) = \text{log}_b (x) + \text{log}_b (y)\)
  • Division Rule: \(\text{log}_b \frac{a}{b} = \text{log}_b (a) - \text{log}_b (b)\)
  • Power Rule: \(\text{log}_b (x^n) = n \times \text{log}_b (x)\)
  • Roots: \(\text{log}_b (x^{1/n}) = \frac{\text{log}_b (x)}{n}\)
Understanding and correctly applying these rules is crucial for simplifying and manipulating complex logarithmic expressions.
In the given exercise, we used the Division Rule, Product Rule, and the property for roots. Mastery of these fundamental rules is the foundation for more advanced logarithmic operations.
Logarithm Simplification
Simplification of logarithmic expressions involves the application of logarithm rules to reduce the expression to its simplest form. Here is a breakdown of the simplification process used in the exercise:
We began with the expression \(\text{log}_4 \frac{\root{4}{z} \times \root{5}{w}}{s^2}\).
  • First, we applied the Division Rule to split the expression into two parts: \(\text{log}_4 (\root{4}{z} \times \root{5}{w}) - \text{log}_4 (s^2)\).
  • Next, we applied the Product Rule to the numerator: \(\text{log}_4 (\root{4}{z}) + \text{log}_4 (\root{5}{w}) - \text{log}_4 (s^2)\).
  • Finally, we simplified the roots using the rule \(\text{log}_b (x^{1/n}) = \frac{\text{log}_b (x)}{n}\): \(\frac{\text{log}_4 (z)}{4} + \frac{\text{log}_4 (w)}{5} - 2 \times \text{log}_4 (s)\).
The result is a simplified expression that uses basic logarithmic properties for easier calculations and understanding.
Logarithmic Expressions
Understanding logarithmic expressions is crucial for working through algebraic problems and equations. These expressions can often look intimidating but can be broken down and simplified using the key logarithmic rules.
Consider the expression in the exercise: \(\text{log}_4 \frac{\root{4}{z} \times \root{5}{w}}{s^2}\).
  • We observe a combination of division, products, and roots.
  • By applying the Division Rule, we separate the numerator from the denominator.
  • Applying the Product Rule allows us to handle the product inside the logarithm.
  • Finally, using the roots property, we deal with the fractional exponents.
This approach demonstrates how effectively using logarithmic properties can simplify even seemingly complex logarithmic expressions. Mastering these techniques can greatly enhance mathematical problem-solving skills.

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Most popular questions from this chapter

A major scientific periodical published an article in 1990 dealing with the problem of global warming. The article was accompanied by a graph that illustrated two possible scenarios. (a) The warming might be modeled by an exponential function of the form $$y=\left(1.046 \times 10^{-38}\right)\left(1.0444^{x}\right)$$ (b) The warming might be modeled by a linear function of the form $$y=0.009 x-17.67$$ In both cases, \(x\) represents the year, and y represents the increase in degrees Celsius due to the warming. Use these functions to approximate the increase in temperature for each of the following years. $$ 2000 $$

Sales (in thousands of units) of a new product are approximated by the function defined by $$ S(t)=100+30 \log _{3}(2 t+1) $$ where \(t\) is the number of years after the product is introduced. (a) What were the sales, to the nearest unit, after 1 yr? (b) What were the sales, to the nearest unit, after 13 yr? (c) Graph \(y=S(t)\)

Solve each equation. Use natural logarithms. When appropriate, give solutions to three decimal places. See Example 2. $$ \ln e^{0.45 x}=\sqrt{7} $$

Match the logarithm in Column I with its value in Column II. (Example: \(\log _{3} 9=2\) because 2 is the exponent to which 3 must be raised in order to obtain 9.) (I) (a) \(\log _{4} 16\) (b) \(\log _{3} 81\) (c) \(\log _{3}\left(\frac{1}{3}\right)\) (d) \(\log _{10} 0.01\) (e) \(\log _{5} \sqrt{5}\) (f) \(\log _{13} 1\) (II) A. \(-2\) B. \(-1\) C. 2 D. 0 E. \(\frac{1}{2}\) F. 4

Solve each equation. Use natural logarithms. When appropriate, give solutions to three decimal places. See Example 2. $$ \ln e^{2 x}=\pi $$
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