Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 10^{x}=50 $$

Step-by-Step Solution

Answer

\( x \approx 1.70 \)

Step 1: Understanding the Equation

We are given the equation \( 10^x = 50 \). Our goal is to solve for the variable \( x \). This means we need to find the value of \( x \) such that the equation holds true.

Step 2: Applying Logarithms

To solve for \( x \), use the property of logarithms that allows us to take the log of both sides. Applying the common logarithm (base 10) to both sides, we get \( \log(10^x) = \log(50) \).

Step 3: Using Logarithm Properties

Using the logarithmic identity \( \log(a^b) = b \cdot \log(a) \), we can simplify the left side of the equation to \( x \cdot \log(10) = \log(50) \). Since \( \log(10) = 1 \), the equation simplifies to \( x = \log(50) \).

Step 4: Calculating the Logarithm

Now compute the value of \( \log(50) \) using a calculator. The approximate value is \( \log(50) \approx 1.69897 \).

Step 5: Rounding to the Nearest Hundredth

The problem asks for the solution to be rounded to the nearest hundredth. Thus, \( x \approx 1.70 \).

Key Concepts

Logarithms

Logarithms are a powerful mathematical tool used to understand and work with exponential relationships. A logarithm answers the question: "To what exponent must the base be raised to produce a specific number?"
For instance, in the equation \( 10^x = 50 \), we use the logarithm to find the exponent \( x \). The common logarithm, which is often denoted as \( \log \), uses 10 as its base. Logarithms simplify working with large numbers and solving equations where the variable is an exponent.

Whenever you encounter an equation like \( a^x = b \), bringing in log helps break it down. By applying the logarithm to both sides of the equation, you transition from the exponential form to a linear form which is much easier to solve.

Properties of Logarithms

The properties of logarithms allow us to manipulate and simplify logarithmic expressions. A fundamental property is the power rule: \( \log(a^b) = b \cdot \log(a) \). This rule helps transform the exponent into a coefficient, making it easier to solve for the variable.

In our example, \( 10^x = 50 \), applying \( \log(10^x) = x \cdot \log(10) \) reduces the equation efficiently. This transformation is crucial because \( \log(10) = 1 \), further simplifying our task to \( x = \log(50) \).

Additional properties include product and quotient rules:

• Product Rule: \( \log(ab) = \log(a) + \log(b) \)
• Quotient Rule: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)

These properties are incredibly useful in handling complex equations by breaking them into simpler parts.

Solving Equations

Solving equations with logarithms involves several systematic steps. Here's how it is typically done:

• Identify and Simplify: Convert the equation to a form suitable for logarithms if necessary, for instance, \( a^x = b \).
• Apply Logarithms: Use logarithms on both sides to handle the variable in the exponent. For our equation, this meant using \( \log(10^x) = \log(50) \).
• Utilize Logarithmic Properties: Simplify further using properties such as the power rule to isolate \( x \).
• Compute and Round: Finally, calculate the logarithmic value using a calculator for precision, and round to the required digit. In this exercise, \( \log(50) \approx 1.69897 \), rounded to \( x \approx 1.70 \).

By following these steps, solving logarithmic equations becomes manageable, allowing you to find the neat values of variables swiftly.