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Chapter 4: Problem 109

a. Evaluate \(\log _{2} 2+\log _{2} 4\) b. Evaluate \(\log _{2}(2 \cdot 4)\) c. How do the values of the expressions in parts (a) and (b) compare?

Short Answer

Expert verified
Both expressions evaluate to 3 and are equal.

Step by step solution

01

Evaluate \(\log _{2} 2\) in part (a)

To evaluate \(\log _{2} 2\), recall that \(\log_b a\) represents the exponent to which the base \(\log b\) must be raised to get \(\log a\). Here, \(\log _{2} 2 = 1\) because \(\2^1 = 2\).
02

Evaluate \(\log _{2} 4\) in part (a)

Next, evaluate \(\log _{2} 4\). Since \(\4 = 2^2\), \(\log _{2} 4 = 2\).
03

Add the results of \(\log _{2} 2\) and \(\log _{2} 4\) in part (a)

Add the results from Steps 1 and 2: \(\log _{2} 2 + \log _{2} 4 = 1 + 2 = 3\).
04

Evaluate \(\log _{2}(2 \cdot 4)\) in part (b)

Evaluate \(\log _{2}(2 \cdot 4)\). Since \(\2 \cdot 4 = 8\), this is equivalent to finding \(\log _{2} 8\). Since \(\8 = 2^3\), \(\log _{2} 8 = 3\).
05

Compare the results of parts (a) and (b)

Both part (a) and part (b) yield the same result, 3. Thus, \(\log_{2} 2 + \log_{2} 4\) is equal to \(\log_{2} (2 \cdot 4)\).

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Key Concepts

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

logarithmic functions
Logarithmic functions are crucial in math, especially when dealing with exponential growth or decay. A logarithm, written as \(\text{log}_b a\), essentially asks: 'To what power must the base (\(b\)) be raised, to produce the number (\(a\))?' For example, \(\text{log}_2 8\) means 'To what power must 2 be raised to yield 8?' The answer is 3 since \2^3 = 8\.

Logarithmic functions have their own unique curve and behavior, distinct from linear or quadratic functions. They tend to rise quickly at first, then level off as the input values get larger. This makes them incredibly useful in describing real-world phenomena like population growth or radioactive decay.
logarithm properties
Understanding the properties of logarithms can help simplify complex equations. These properties include:
  • Product Rule: \(\text{log}_b(MN) = \text{log}_bM + \text{log}_bN\)
  • Quotient Rule: \(\text{log}_b(M/N) = \text{log}_bM - \text{log}_bN\)
  • Power Rule: \(\text{log}_b(M^k) = k \text{log}_bM\)

These rules transform complicated logarithmic expressions into simpler ones. For instance, consider the product rule. In the exercise, \(\text{log}_2 2 + \text{log}_2 4\) can be simplified using the product rule to \(\text{log}_2 (2 \times 4) = \text{log}_2 8 = 3\), showing how crucial these properties are for simplifying calculations.
base and exponent relationships
The relationship between bases and exponents is at the core of logarithms. If you understand how raising a base to a power works, you can better grasp logarithms. For example, take 2 raised to the power of 3: \2^3 = 8\. Here, 2 is the base, and 3 is the exponent.

The logarithm \(\text{log}_2 8\) asks: 'What exponent should 2 be raised to, to get 8?' The answer is, of course, 3. This relationship helps in solving various logarithmic problems by converting them into simpler exponential forms.
logarithm addition
Adding logarithms might seem challenging at first, but it's straightforward with the right knowledge. According to the product rule of logarithms: \( \text{log}_b(M) + \text{log}_b(N) = \text{log}_b(M \times N) \).

In the given exercise, the expression \( \text{log}_2 2 + \text{log}_2 4 \) is equal to \( \text{log}_2 (2 \times 4) \). So, \( \text{log}_2 8 = 3 \). This addition rule simplifies calculations and helps solve logarithmic equations more efficiently.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{4}(3 d)+\log _{4} 1=\log _{4}(3 d) $$

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{8} 5 $$

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \ln 10=\frac{1}{\log e} $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(7^{4 x-1}=3^{5 x}\)

The isotope of plutonium \({ }^{238} \mathrm{Pu}\) is used to make thermoelectric power sources for spacecraft. Suppose that a space probe is launched in 2012 with \(2.0 \mathrm{~kg}\) of \({ }^{238} \mathrm{Pu}\) a. If the half-life of \({ }^{238} \mathrm{Pu}\) is \(87.7 \mathrm{yr}\), write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the quantity \(Q(t)\) of \({ }^{238} \mathrm{Pu}\) left after \(t\) years. b. If \(1.6 \mathrm{~kg}\) of \({ }^{238} \mathrm{Pu}\) is required to power the spacecraft's data transmitter, for how long will scientists be able to receive data? Round to the nearest year.
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