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Chapter 4: Problem 57

Write the logarithmic expression as a single logarithm with coefficient \(1,\) and simplify as much as possible. (See Exercises \(5-6)\) $$ 4 \log _{8} m-3 \log _{8} n-2 \log _{8} p $$

Short Answer

Expert verified
\( \log_{8} \left(\frac{m^4}{n^3 p^2}\right) \)

Step by step solution

01

- Use the power rule for logarithms

The power rule of logarithms states that \[ k \, \text{log}_a(b) = \text{log}_a(b^k) \] Use this rule to rewrite each term with the coefficients as exponents inside the logarithms\[ 4 \log_{8} m = \log_{8} (m^4) \]\[ 3 \log_{8} n = \log_{8} (n^3) \]\[ 2 \log_{8} p = \log_{8} (p^2) \]
02

- Substitute the exponents back into the logarithmic expressions

Replace each term with its equivalent using the exponents:\[ \log_{8} (m^4) - \log_{8} (n^3) - \log_{8} (p^2) \]
03

- Use the properties of logarithms for subtraction

The property of logarithms states that \[ \log_{a} \left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c) \] Apply this property step-by-step to combine the terms:\[ \log_{8} (m^4) - \log_{8} (n^3) = \log_{8} \left(\frac{m^4}{n^3}\right) \]Then, apply it again:\[ \log_{8} \left(\frac{m^4}{n^3}\right) - \log_{8} (p^2) = \log_{8} \left(\frac{m^4}{n^3 p^2}\right) \]
04

- Simplify the final expression

Since all terms are inside a single logarithm now, it is simplified as much as possible. The final expression is \[ \log_{8} \left(\frac{m^4}{n^3 p^2}\right) \]

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Key Concepts

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

power rule for logarithms
The power rule for logarithms is an essential property that helps in simplifying logarithmic expressions. It states that multiplying a logarithm by a coefficient is equivalent to raising the argument of the logarithm to the power of that coefficient. Mathematically, this can be written as: \[ k \, \text{log}_a(b) = \text{log}_a(b^k) \] Here's how you apply it to the given expression:
  • For \(4 \, \log_8 m\), the coefficient 4 moves inside the logarithm, becoming \( \log_8 (m^4) \).
  • For \(3 \, \log_8 n\), the coefficient 3 becomes \( \log_8 (n^3) \).
  • For \(2 \, \log_8 p\), the coefficient 2 translates to \( \log_8 (p^2) \).
This step transforms the original expression into: \[ \log_8 (m^4) - \log_8 (n^3) - \log_8 (p^2) \]
properties of logarithms
To simplify logarithmic expressions, knowing the properties of logarithms is crucial. One important property is the subtraction rule: \[ \log_a \left( \frac{b}{c} \right) = \log_a(b) - \log_a(c) \] This property tells us that the subtraction of logarithms translates to the division of their arguments. In the given problem, we apply this rule in two steps:
  • First, combine \( \log_8 (m^4) \) and \( - \log_8 (n^3) \) into \( \log_8 \left( \frac{m^4}{n^3} \right) \).
  • Next, combine \( \log_8 \left( \frac{m^4}{n^3} \right) \) and \( - \log_8 (p^2) \) into \( \log_8 \left( \frac{m^4}{n^3 p^2} \right).\)
Applying these properties simplify the mixture of logarithmic terms into a single, more manageable expression.
logarithmic simplification
Logarithmic simplification involves using properties and rules to combine multiple logarithmic terms into a single log term. In this exercise, we start with: \[ 4 \log_8 m - 3 \log_8 n - 2 \log_8 p \] By applying the power rule and properties of logarithms, we reduce the expression to: \[ \log_8 \left( \frac{m^4}{n^3 \cdot p^2} \right) \] Simplifying logarithmic expressions typically involves:
  • Moving coefficients inside as exponents using the power rule.
  • Combining logarithmic terms via addition and subtraction properties.
  • Ensuring the final expression is in its simplest form by combining all terms into one logarithm.
In this case, the final simplified expression has no further simplifications possible, presenting the most simplified and concise form of the given logarithmic expression.

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Most popular questions from this chapter

Show that \(\left(\frac{e^{x}+e^{-x}}{2}\right)^{2}-\left(\frac{e^{x}-e^{-x}}{2}\right)^{2}=1\).

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The number of computers \(N(t)\) (in millions) infected by a computer virus can be approximated by $$N(t)=\frac{2.4}{1+15 e^{-0.72 t}}$$ where \(t\) is the time in months after the virus was first detected. a. Determine the number of computers initially infected when the virus was first detected. b. How many computers were infected after 6 months? Round to the nearest hundred thousand. c. Determine the amount of time required after initial detection for the virus to affect 1 million computers. Round to 1 decimal place. d. What is the limiting value of the number of computers infected according to this model?

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