Problem 52 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The solution is \( c > 256 \).

Step 1: 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.

Step 2: 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 \( c > 2^8 \).

Step 3: 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 \( c > 256 \).

Step 4: 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.

Key Concepts

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’s 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.