Problem 54 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. \(\log _{\frac{1}{3}} p<0\)

Step-by-Step Solution

Answer

The solution is \( p > 1 \).

Step 1: Understand the Inequality

We are solving the inequality \( \log_{\frac{1}{3}} p < 0 \). This means we want the power to which we raise \( \frac{1}{3} \) to get \( p \) to be less than zero.

Step 2: Know the Base Properties

Since the base \( \frac{1}{3} \) is less than 1, if \( p \) is greater than 1, then \( \log_{\frac{1}{3}} p \) will be negative. This is because raising a fraction less than one to a negative power results in a number greater than one.

Step 3: Write the Inequality Condition

For \( \log_{\frac{1}{3}} p < 0 \), we can deduce that \( p > 1 \). This is because the base fraction, \( \frac{1}{3} \), raised to a negative number results in a number greater than 1.

Step 4: Check the Solution

To verify, pick a value \( p = 2 \). We know \( \frac{1}{3}^{-1} = 3 \), which is true, so \( \log_{\frac{1}{3}} 2 < 0 \) is correct. Thus, \( p > 1 \) satisfies our original inequality.

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is crucial when solving logarithmic equations and inequalities. Logarithms help us determine the power to which a specific base must be raised in order to obtain a given number. Here, we are dealing with the base \(\frac{1}{3}\). Logarithmic expressions follow certain rules:

• The logarithm of a number greater than 1 with a fraction as a base (less than 1) produces a negative result.
• The identity \(\log_b(b) = 1\) holds true for any base \(b\), meaning that raising the base to the first power returns the base.
• The logarithm of 1 in any base is always 0: \(\log_b(1) = 0\).

In our exercise, understanding these properties helps us narrow down the condition where \(\log_{\frac{1}{3}} p < 0\). The power (or exponent) must be negative for \(\frac{1}{3}\) to result in a number greater than \(p\).

Inequality Solving

When it comes to solving inequalities involving logarithms, we need to understand how the properties of the base affect the inequality. Given the inequality \(\log_{\frac{1}{3}} p < 0\), our goal is to find the values of \(p\) that satisfy this condition.- A fractional base less than 1 causes the inequality symbol to flip when converting to exponential form.- For \(p\) to make \(\log_{\frac{1}{3}} p\) negative, \(p\) must be a number greater than what the base raised to a positive power would produce.Thus, solving \(\log_{\frac{1}{3}} p < 0\) results in the condition \(p > 1\). This is derived from the fact that a higher positive exponent on a fractional base yields a smaller number, hence a negative logarithm when \(p > 1\).

Fractional Bases

Using fractional bases in logarithmic equations can at first seem tricky due to their inversion of the usual exponent rule.A fractional base like \(\frac{1}{3}\) changes the usual behavior:

• Exponents that are positive result in smaller outputs, compared to non-fractional bases.
• If an exponent is negative, the base \(\frac{1}{3}\) results in a larger value, effectively inverting the relationship.
• Therefore, for \(\log_{\frac{1}{3}} p < 0\), we look for conditions where the exponent is negative enough to produce a result greater than 1.

This shows why, in reversing the inequality and exponent rule, \(p\) needs to be greater than 1, ensuring that \(\log_{\frac{1}{3}} p\) is indeed negative.