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Chapter 9: Problem 52

Solve each equation or inequality. Check your solutions. \(\log _{2} c>8\)

Short Answer

Expert verified
The solution is \( c > 256 \).

Step by step solution

01

Understand the Inequality

The given inequality is \( \log_{2} c > 8 \). This means that the base-2 logarithm of \( c \) must be greater than 8. We want to find the range of values for \( c \) that satisfy this inequality.
02

Rewrite the Logarithmic Inequality

To solve the inequality \( \log_{2} c > 8 \), we first convert it into its exponential form. The inequality \( \log_{b} x > n \) can be rewritten as \( x > b^n \). Therefore, our inequality becomes \( c > 2^8 \).
03

Calculate the Exponentiation

We need to determine the value of \( 2^8 \). Calculating this gives \( 2^8 = 256 \). Therefore, the inequality \( c > 2^8 \) becomes \( c > 256 \).
04

Verify the Solution

To verify, check a value greater than 256, say \( c = 300 \). Calculating \( \log_{2} 300 \) is approximately \( 8.23 \), which is greater than 8. Therefore, this satisfies the original inequality. On the other hand, a value such as \( c = 250 \) gives \( \log_{2} 250 \approx 7.96 \), which does not satisfy the inequality.

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Key Concepts

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

Exponential Form
The exponential form is a way to express the solution or transformation of a logarithm. Basically, switching from a logarithmic form to an exponential form simplifies solving the equation or inequality. To convert a logarithmic expression \( \log_{b} x > n \) into exponential form, we rewrite it as \( x > b^n \).
  • This transformation is useful because exponentiation often feels more straightforward, enabling easier computation and interpretation.
  • In the case of our inequality \( \log_{2} c > 8 \), translating it to exponential form makes it easier to identify that \( c \) must be greater than \( 2^8 \), which is 256.
  • By converting to exponential form, we're stripping away the complexity of the logarithm, helping make inequalities such as this much more intuitive to solve.
Inequality Solving
Solving inequalities involves finding a range of possible solutions that satisfy a given condition. When you're dealing with logarithmic inequalities, the approach revolves around transforming the inequality into a practical form you can solve. Here鈥檚 how it works:
  • First, understand the inequality in question. For instance, \( \log_{2} c > 8 \) means finding values of \( c \) where its logarithm, base 2, is indeed greater than 8.
  • Next, we convert it to exponential form: \( c > 2^8 \).
  • Then, evaluating \( 2^8 \) helps to give our inequality a tangible boundary. Here, \( 2^8 \) equals 256.
  • Thus, this inequality-solving process shows that \( c \) must be more than 256 to satisfy the inequality.
Check your solutions by inserting likely values into the original inequality. Evaluating, for example, \( c = 300 \) (which yields a value greater than 8 in logarithmic terms) ensures your understanding is thorough.
Base-2 Logarithm
A base-2 logarithm, also known as binary logarithm, is a logarithm where the base is 2. It expresses how many times the base must be multiplied by itself to reach a given number.
  • The expression \( \log_{2} c \) asks, "To what power must 2 be raised to obtain \( c \)?"
  • Base-2 logarithms are particularly common in computer science because they relate to binary systems.
  • In the log inequality \( \log_{2} c > 8 \), you're seeking numbers \( c \) requiring elevation of 2 to more than the 8th power to achieve them.
Understanding base-2 logarithms lays a helpful foundation in both mathematical contexts and beyond, equipping you with vital problem-solving tools in fields like cryptography, networking, and algorithm design.

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Most popular questions from this chapter

PREREQUISITE SKILL Solve each equation or inequality. Check your solutions. $$ \log _{2} 3 x>\log _{2} 5 $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln (x-7)=2\)

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REASONING Use the properties of exponents to prove the Quotient Property of Logarithms.

In 2005 , the world's population was about 6.5 billion. If the world's population continues to grow at a constant rate, the future population \(P,\) in billions, can be predicted by \(P=6.5 e^{0.02 t},\) where \(t\) is the time in years since 2005. Some experts have estimated that the world鈥檚 food supply can support a population of at most 18 billion. According to this model, for how many more years will the food supply be able to support the trend in world population growth?
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