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Chapter 8: Problem 22

If \(\log _{3} m=n,\) then determine \(\log _{3} m^{4},\) in terms of \(n\).

Short Answer

Expert verified
\( \log_{3} m^{4} = 4n \)

Step by step solution

01

- Understand the given information

We are given \( \log _{3} m=n. \) This means that when we raise 3 to the power of n, we get m, i.e., \( \3^{n} = m. \)
02

- Use the logarithm power rule

The logarithm power rule tells us that \( \log_b (a^c) = c \log_b (a). \) We want to find \( \log_{3} m^{4}. \)
03

- Apply the power rule to the given logarithm

Using the power rule, we rewrite \( \log_{3} m^{4} \) as follows: \( \log_{3} m^{4} = 4 \log_{3} m. \)
04

- Substitute the given value

From the given information, we have \( \log_{3} m = n. \) Substituting this in, we get \( \log_{3} m^{4} = 4n. \)

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Key Concepts

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

Logarithm Power Rule
The logarithm power rule is a handy property when dealing with powers inside logarithms. This rule states that \[ \log_b (a^c) = c \log_b (a). \] It lets us move the exponent in front of the log, making the math simpler.
For example, let's say you have \( \log_3 (m^4). \) According to the power rule, you can rewrite it as \( 4 \log_3 (m). \)
Instead of dealing with \( m^4, \) you now only need to handle \( 4 \log_3(m). \) The rule is very helpful and is widely used in solving logarithmic problems.
Always remember this rule, as it simplifies complex expressions and solves issues faster. Understanding it deeply ensures you won't make mistakes when you apply it.
Logarithmic Properties
Understanding the properties of logarithms is essential. Let's look at the main ones:
  • Product Rule: \( \log_b (xy) = \log_b (x) + \log_b (y) \) - This splits the log of a product into a sum of logs.

  • Quotient Rule: \( \log_b (\frac{x}{y}) = \log_b (x) - \log_b (y) \) - This splits the log of a quotient into a difference of logs.

  • Change of Base Formula: \( \log_b (a) = \frac{\log_c (a)}{\log_c (b)} \) - Changes the base of a logarithm.

  • Power Rule: \( \log_b (a^c) = c \log_b (a) \) - We have detailed this one more in the previous section.

By mastering these properties, you can handle almost any logarithmic problem with ease. Review these and make sure you understand how to apply them.
Precalculus Problem Solving
Precalculus often involves manipulating and simplifying expressions to solve problems. For logarithms, it means using important properties to rewrite and solve log equations.
Let's apply this to a sample problem. Say you are given \( \log_3 m = n \) and asked to find \( \log_3 (m^4) \).
Step by step:
  • Identify given information: \( \log_3 m = n, \) which implies \( 3^n = m. \)

  • Use the power rule: Rewriting \( \log_3 (m^4) \) gives you \( 4 \log_3 (m). \)

  • Substitute: Given \( \log_3 m = n, \) replace it in the equation to get \( 4n. \)

Breaking it down into smaller steps makes the problem easier to handle. This is key in all precalculus problem solving. Familiarize yourself with fundamental rules and apply them methodically.

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Most popular questions from this chapter

Determine the equation of the transformed image after the transformations described are applied to the given graph. a) The graph of \(y=2 \log _{5} x-7\) is reflected in the \(x\) -axis and translated 6 units up. b) The graph of \(y=\log (6(x-3))\) is stretched horizontally about the \(y\) -axis by a factor of 3 and translated 9 units left.

Decide whether each equation is true or false. Justify your answer. Assume \(c, x,\) and \(y\) are positive real numbers and \(c \neq 1\). a) \(\frac{\log _{e} x}{\log _{e} y}=\log _{e} x-\log _{e} y\) b) \(\log _{c}(x+y)=\log _{c} x+\log _{c} y\) c) \(\log _{c} c^{n}=n\) d) \(\left(\log _{c} x\right)^{n}=n \log _{c} x\) e) \(-\log _{c}\left(\frac{1}{x}\right)=\log _{c} x\)

Write each expression as a single logarithm in simplest form. a) \(\log _{0} x-\log _{9} y+4 \log _{9} z\) b) \(\frac{\log _{3} x}{2}-2 \log _{3} y\) c) \(\log _{6} x-\frac{1}{5}\left(\log _{6} x+2 \log _{6} y\right)\) d) \(\frac{\log x}{3}+\frac{\log y}{3}\)

The Palermo Technical Impact Hazard scale was developed to rate the potential hazard impact of a near-Earth object. The Palermo scale, \(P,\) is defined as \(P=\log R\) where \(R\) is the relative risk. Compare the relative risks of two asteroids, one with a Palermo scale value of -1.66 and the other with a Palermo scale value of -4.83.

a) If \(g(x)=\log _{\frac{1}{4}} x,\) state the equation of the inverse, \(g^{-1}(x)\) b) Sketch the graph of \(g(x)\) and its inverse. Identify the following characteristics of the inverse graph: . the domain and range \cdot the \(x\) -intercept, if it exists \cdot the \(y\) -intercept, if it exists \bullet the equations of any asymptotes
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