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Chapter 5: Problem 4

Express as a sum of logarithms and simplify, if possible. $$\log _{4}(64 \cdot 4)$$

Short Answer

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4

Step by step solution

01

Apply the product rule

Use the properties of logarithms, specifically the product rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\). In this case, \(\log_{4}(64 \cdot 4)\) can be expressed as \(\log_4(64) + \log_4(4)\).
02

Simplify individual logarithms

Now simplify each logarithm separately. Recognize that \(64 = 4^3\), so \(\log_4(64) = \log_4(4^3) \). Using the power rule \(\log_b(x^k) = k \log_b(x)\), we get \(\log_4(64) = 3 \log_4(4)\). Furthermore, \(\log_4(4) = 1\) because any logarithm of a base to itself is 1.
03

Combine the simplified terms

Substitute the simplified value back into the expression: \(\log_4(64) + \log_4(4) = 3 \log_4(4) + \log_4(4)\). This simplifies to \(3 \cdot 1 + 1 = 3 + 1 = 4\).

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Key Concepts

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

Product Rule
The product rule for logarithms is a fundamental property that makes it easier to work with logarithms when you have products of numbers. According to the product rule, \(\log_b(xy) = \log_b(x) + \log_b(y)\). This means the logarithm of a product is the same as the sum of the logarithms of each individual factor.
To illustrate this, let's take the example from the exercise: \(\log_{4}(64 \cdot 4)\). Using the product rule, we can break this into \(\log_{4}(64) + \log_{4}(4)\). This simplification allows us to handle each part separately, making the process more manageable.
Power Rule
The power rule is another crucial property of logarithms, which states that \(\log_b(x^k) = k \log_b(x)\). This tells us that the logarithm of a number raised to an exponent is equal to that exponent multiplied by the logarithm of the base number.
Returning to our exercise, we need to simplify \(\log_4(64)\). We recognize that 64 can be written as \(4^3\). Thus, \(\log_4(64) = \log_4(4^3)\). Using the power rule, this becomes \(3 \log_4(4)\). Given that \(\log_4(4) = 1\), it simplifies further to \(3 \cdot 1 = 3\).
Logarithm Properties
Understanding the properties of logarithms is essential for simplifying and manipulating logarithmic expressions. Some key properties include:
  • Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\).
  • Power Rule: \(\log_b(x^k) = k \log_b(x)\).
  • Logarithm of the base to itself: \(\log_b(b) = 1\).

These properties were used in the exercise to break down the expression \(\log_{4}(64 \cdot 4)\). We first applied the product rule: \(\log_4(64) + \log_4(4)\). Then, we used the power rule to simplify \(\log_4(64)\) to \(3 \log_4(4)\). Finally, knowing that \(\log_4(4) = 1\), we combined the simplified terms to reach the final answer: 4.

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Most popular questions from this chapter

Growth of a Stock. The value of a stock is given by the function $$ V(t)=58\left(1-e^{-1.1 t}\right)+20 $$ where \(V\) is the value of the stock after time \(t,\) in months. a) Graph the function. b) Find \(V(1), V(2), V(4), V(6),\) and \(V(12)\) c) After how long will the value of the stock be \(\$ 75 ?\)

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{5}(x+4)+\log _{5}(x-4)=2$$

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) Which graphs open up?

Use a graphing calculator to find the approximate solutions of the equation. $$2^{x}-5=3 x+1$$

Price of Admission to the Magic Kingdom. In \(2015,\) the price of a one-day, one-park admission to Disney's Magic Kingdom in Florida rose to \(\$ 105 .\) The exponential function $$ D(x)=4.532(1.078)^{x} $$ where \(x\) is the number of years after \(1971,\) models the price of a ticket. (Source: AllEars.net, an independent Disney consumer website) Find the price of a ticket in \(1980,\) in \(2000,\) and in \(2012 .\) Then use the function to project the price of a ticket in 2020 .
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