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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

Solve for \(x\) a) \(2 \log _{3} x=\log _{3} 32+\log _{3} 2\) b) \(\frac{3}{2} \log _{7} x=\log _{7} 125\) c) \(\log _{2} x-\log _{2} 3=5\) d) \(\log _{6} x=2-\log _{6} 4\)

The growth of a new social networking site can be modelled by the exponential function \(N(t)=1.1^{t},\) where \(N\) is the number of users after \(t\) days. a) Write the equation of the inverse. b) How long will it take, to the nearest day, for the number of users to exceed \(1 000 000 ?\)

The apparent magnitude of a celestial object is how bright it appears from Earth. The absolute magnitude is its brightness as it would seem from a reference distance of 10 parsecs (pc). The difference between the apparent magnitude, \(m\), and the absolute magnitude, \(M,\) of a celestial object can be found using the equation \(m-M=5 \log d-5,\) where \(d\) is the distance to the celestial object, in parsecs. Sirius, the brightest star visible at night, has an apparent magnitude of -1.44 and an absolute magnitude of 1.45 a) How far is Sirius from Earth in parsecs? b) Given that \(1 \mathrm{pc}\) is approximately 3.26 light years, what is the distance in part a) in light years?

The graph of \(y=\log _{3} x\) has been transformed to \(y=a \log _{3}(b(x-h))+k\) Find the values of \(a, b, h,\) and \(k\) for each set of transformations. Write the equation of the transformed function. a) a reflection in the \(x\) -axis and a translation of 6 units left and 3 units up b) a vertical stretch by a factor of 5 about the \(x\) -axis and a horizontal stretch about the \(y\) -axis by a factor of \(\frac{1}{3}\). c) a vertical stretch about the \(x\) -axis by a factor of \(\frac{3}{4},\) a horizontal stretch about the \(y\) -axis by a factor of \(4,\) a reflection in the \(y\) -axis, and a translation of 2 units right and 5 units down.

The formula for the Richter magnitude, \(M\) of an earthquake is \(M=\log \frac{A}{A_{0}},\) where \(A\) is the amplitude of the ground motion and \(A_{0}\) is the amplitude of a standard earthquake. In \(1985,\) an earthquake with magnitude 6.9 on the Richter scale was recorded in the Nahanni region of the Northwest Territories. The largest recorded earthquake in Saskatchewan occurred in 1982 near the town of Big Beaver. It had a magnitude of 3.9 on the Richter scale. How many times as great as the seismic shaking of the Saskatchewan earthquake was that of the Nahanni earthquake?
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