Problem 18 - Algebra 2

Question

Solve each inequality. Check your solutions. \(\log _{16} x \geq \frac{1}{4}\)

Step-by-Step Solution

Answer

The solution is \( x \geq 2 \).

Step 1: Rewrite the Inequality Using Exponential Form

The inequality involves a logarithm, so let's convert it to its exponential form. The expression \( \log_{16} x \geq \frac{1}{4} \) can be rewritten as an exponential inequality: \( x \geq 16^{\frac{1}{4}} \).

Step 2: Simplify the Exponential Expression

Now, simplify \( 16^{\frac{1}{4}} \). The base 16 can be rewritten as \( 2^4 \), so \( 16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \). Thus, we have \( x \geq 2 \).

Step 3: Check the Solution

To verify our solution of \( x \geq 2 \), substitute \( x = 2 \) into the original inequality: \( \log_{16} 2\). Calculate \( \log_{16} 2 \) using the change of base formula: \[ \log_{16} 2 = \frac{\log_{10} 2}{\log_{10} 16} \].\Approximating using calculator values, \( \log_{10} 2 \approx 0.3010 \) and \( \log_{10} 16 \approx 1.2041 \), giving \[ \log_{16} 2 \approx \frac{0.3010}{1.2041} \approx 0.25 \]. This confirms that \( 0.25 \geq \frac{1}{4} \) is true, so \( x = 2 \) satisfies the inequality.

Key Concepts

Logarithms

Logarithms are mathematical operations that help us solve equations involving exponential growth or decay. They are the inverse of exponentiation.
In simpler terms, if you know the result of a power and the base, a logarithm helps you determine the exponent.
Here are some important points about logarithms:

• The logarithm of a number is the exponent to which the base must be raised to get that number. If \( b^y = x \), then \( \log_b(x) = y \).
• Common bases for logarithms include 10 (common logarithm) and \( e \) (natural logarithm).
• In the inequality \( \log_{16}(x) \geq \frac{1}{4} \), we want to find out when the logarithm of \( x \) to the base 16 is greater than or equal to 1/4.

Understanding logarithms is crucial, as they allow us to simplify complex calculations and solve for variables in exponential equations.

Exponential Expressions

Exponential expressions involve numbers raised to a power or exponent. They are used in various fields to model growth or decay phenomena.

• An expression like \( a^b \) consists of a base \( a \) and an exponent \( b \).
• The base indicates the number that is being multiplied, and the exponent tells you how many times to multiply the base by itself.

In the exercise, we converted a logarithmic inequality into an exponential form: \( x \geq 16^{\frac{1}{4}} \).
This step simplified the expression to make it easier to solve.
By changing \( 16 \) into \( 2^4 \), the expression \( 16^{\frac{1}{4}} \) became \( 2^{1} \).
Therefore, the inequality became \( x \geq 2 \), making the solution straightforward.

Change of Base Formula

The Change of Base Formula is a handy tool for evaluating logarithms that are not based on common numbers like 10 or \( e \).
It allows us to rewrite a logarithm in terms of logarithms with a different base, typically base 10, for easier computation.

• The formula is \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), where \( k \) is a new base of choice.

In our exercise, we needed the value of \( \log_{16}(2) \). By applying the change of base formula, it was computed as \[ \log_{16}(2) = \frac{\log_{10}(2)}{\log_{10}(16)} \].
This conversion made the calculation more straightforward, especially with a calculator.
Understanding and using the Change of Base Formula simplifies problems involving non-standard bases.