Problem 31 - Algebra 2

Question

Solve each equation or inequality. Round to four decimal places. $$ 2^{3 p}>1000 $$

Step-by-Step Solution

Answer

The solution is \(p > 3.3190\).

Step 1: Understand the Problem

We are tasked with solving the inequality \(2^{3p} > 1000\). This means we need to find the values of \(p\) that make this inequality true.

Step 2: Apply Logarithms

To solve this exponential inequality, we take the logarithm of both sides. This will allow us to bring the variable \(p\) out of the exponent.\[\log(2^{3p}) > \log(1000)\]

Step 3: Simplify Using Logarithmic Properties

Using the property of logarithms \(\log(a^b) = b\log(a)\), we can simplify the inequality:\[3p \cdot \log(2) > \log(1000)\]

Step 4: Solve for p

Divide both sides by \(3 \cdot \log(2)\) to solve for \(p\):\[p > \frac{\log(1000)}{3 \cdot \log(2)}\]

Step 5: Compute the Numerical Value

Calculate the value by substituting \(\log(1000)\) and \(\log(2)\) with their approximate values:\[p > \frac{3 \cdot \log(10)}{3 \cdot 0.3010} = \frac{3}{0.903}\]Calculate \(p\) after simplifying further:\[p > 3.3190\]

Key Concepts

Logarithmic Properties

Logarithmic properties are essential when dealing with exponential equations and inequalities. They help simplify the math involved. For instance, one useful property is that the logarithm of a power allows us to move the exponent in front of the log. This turns a multiplication into a simpler multiplication problem. We use the rule:

• \( \log(a^b) = b \cdot \log(a) \)

In our exercise, this property allows us to handle the expression \( 2^{3p} \) by rewriting it as \( 3p \cdot \log(2) \). This is a key step because it moves the variable \( p \) out of the exponent, making it much easier to solve for \( p \).
Another helpful property is that logarithms convert multiplication and division into addition and subtraction. Logarithms also have specific bases, and understanding which base you are dealing with (common logarithm with base 10 in this exercise) allows you to use accurate log values from a calculator.
Recognizing and applying these properties effectively simplifies solving exponential inequalities.

Solving Inequalities

Inequalities, unlike equations, tell us that one side is larger or smaller than the other. Solving inequalities involves finding a set of values that make the inequality true.
In our example, we have the inequality \( 2^{3p} > 1000 \), and we want to find all \( p \) values that satisfy this condition. We approach these inequalities by:

• First transforming the exponential inequality using logarithms, which converts it into a linear inequality.
• Next, simplifying it using logarithmic properties and isolating the variable \( p \).
• Finally, computing the range for the variable \( p \).

Through these steps, we transition from an exponential form to a linear form, which is much easier to solve. Remember, solving inequalities is not just about finding a single solution but understanding the range of possible solutions that satisfy the inequality condition. It's essential to perform each step carefully to achieve the correct solution set.

Exponent Rules

Exponent rules govern how we handle expressions with powers and are fundamental in simplifying and solving exponential equations and inequalities.
Important rules related to exponents include:

• Multiplication of the same bases: \( a^m \cdot a^n = a^{m+n} \)
• Power of a power: \( (a^m)^n = a^{m \cdot n} \)
• Division of same bases: \( \frac{a^m}{a^n} = a^{m-n} \)

In the original problem, understanding exponents helps when dealing with the expression \( 2^{3p} \). Here, \( p \) is multiplied by the power 3, illustrating the ʻPower of a Powerʼ rule when transformed using logarithms.
By adhering to these rules, we can manipulate expressions into simpler forms. This simplifies the overall solving process and makes it feasible to tackle both simple and complex inequality problems effectively.