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Chapter 3: Problem 24

In Exercises 23 - 28, use the properties of logarithms to rewrite and simplify the logarithmic expression. \( \log_2 \left(4^2 \cdot 3^4\right) \)

Short Answer

Expert verified
The simplified form of \( \log_2 \left(4^2 \cdot 3^4\right) \) is \(4 + 4 \cdot log_2(3)\).

Step by step solution

01

Apply the product property of logarithms.

Rewrite the logarithmic expression using the rule \(log_b(M \cdot N) = log_b(M) + log_b(N)\), so we get \(log_2(4^2) + log_2(3^4)\).
02

Apply the power property of logarithms.

Use the property that says \(log_b(a^k) = k \cdot log_b(a)\). After applying on our expression, the result will be \(2 \cdot log_2(4) + 4 \cdot log_2(3)\).
03

Evaluate \(\log_2 (4)\) and \(\log_2 (3)\).

The logarithm base 2 of 4 is 2 (because \(2^2 = 4\)) and for base 2 of 3 we have an irrational number, which can be approximated or left as it is. We get: \(2 \cdot 2 + 4 \cdot log_2(3)\).
04

Simplify the expression.

Finally, perform the multiplication to arrive at the simplified expression: \(4 + 4 \cdot log_2(3)\).

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Key Concepts

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

Product Property of Logarithms
When dealing with logarithmic expressions, the product property is your friend in simplifying complex scenarios. This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, this is written as:
  • \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \)
In our example, \( \log_2(4^2 \cdot 3^4) \), we apply this property to split the expression into two simpler parts:
  • \( \log_2(4^2) + \log_2(3^4) \)
This step is crucial because it helps break down the complexity, making further simplification easier. By treating each factor separately, it becomes more manageable to apply additional logarithmic properties.
Power Property of Logarithms
After breaking down a logarithmic expression using the product rule, the power property of logarithms can further simplify the problem. This property allows you to move exponents to the front of the logarithm as a coefficient. It's expressed by the formula:
  • \( \log_b(a^k) = k \cdot \log_b(a) \)
Applying this to each term, you transform:
  • \( \log_2(4^2) \) into \( 2 \cdot \log_2(4) \)
  • \( \log_2(3^4) \) into \( 4 \cdot \log_2(3) \)
With the exponents moved outside, it's much easier to calculate or estimate the value of each logarithm, leading you closer to a simplified form of the expression.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves applying properties in a coherent sequence to achieve the most fundamental form. Once you've dealt with the structure using properties, you perform calculations or approximate if needed. In our example, computing \( \log_2(4) \) gives 2, because \( 2^2 = 4 \). For \( \log_2(3) \), it鈥檚 irrational, but you can use approximations. The expression simplifies to:
  • \( 2 \cdot 2 + 4 \cdot \log_2(3) \)
  • Resulting in \( 4 + 4 \cdot \log_2(3) \)
This final step can often leave terms with logarithms if the value is irrational. But by reducing unnecessary complexity, the expression becomes more digestible, practical for applications, or ready for further steps.

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Most popular questions from this chapter

The numbers of freestanding ambulatory care surgery centers in the United States from \( 2000 \) through \( 2007 \) can be modeled by \( y = 2875 + \dfrac{2635.11}{1 + 14.215e^{0.8038t^,}} 0 \le t \le 7 \) where \( t \) represents the year, with \( t = 0 \) corresponding to \( 2000 \).(Source: Verispan) (a) Use a graphing utility to graph the model. (b) Use the trace feature of the graphing utility to estimate the year in which the number of surgery centers exceeded \( 3600 \).

The amount of time (in hours per week)a student utilizes a math-tutoring center roughly follows the normal distribution \( y = 0.7979e^{-(x - 5.4)^2/0.5} \), \( 4 \le x \le 7 \), where \( x \) the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.

In Exercises 25 and 26, determine the time necessary for \( \$1000 \) to double if it is invested at interest rate \( r \) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. \( r = 6.5\% \)

In Exercises 23 and 24, determine the principal \( P \) that must be invested at rate \( r \) compounded monthly, so that \( \$500,000 \) will be available for retirement in \( t \) years. \( r = 3\dfrac{1}{2}\% \), \( t = 15 \)

In Exercises 65 - 68, use the following information for determining sound intensity. The level of sound \( \beta \), in decibels, with an intensity of \( I \), is given by \( \beta = 10 \log\left(I/I_0\right) \), where \( I_0 \) is an intensity of \( 10^{-12} \) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 65 and 66, find the level of sound \( \beta \) (a) \( I = 10^{-11} \) watt per \( m^2 \) (rustle of leaves) (b) \( I = 10^2 \) watt per \( m^2 \) (jet at 30 meters) (c) \( I = 10^{-4} \) watt per \( m^2 \) (door slamming) (d) \( I = 10^{-2} \) watt per \( m^2 \) (siren at 30 meters)
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