Problem 85 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \log x=6$$

Step-by-Step Solution

Answer

x = 1000

Step 1: Understand the Equation

The equation given is \( 2 \log x = 6 \). This is a logarithmic equation where we need to solve for \( x \).

Step 2: Divide Both Sides by 2

To isolate the logarithm, divide both sides of the equation by 2: \[ \log x = \frac{6}{2} \] After performing the division, we have: \[ \log x = 3 \]

Step 3: Use Exponential Form

The equation \( \log x = 3 \) can be rewritten in its exponential form. By the definition of logarithms: \[ x = 10^3 \] This converts the equation from logarithmic form to exponential form, assuming the base is 10 (common logarithm).

Step 4: Calculate the Power

Calculate \( 10^3 \): \[ x = 1000 \] This calculation provides the solution for \( x \).

Key Concepts

Solve for x

We begin by addressing the core task of solving for \( x \). Given a logarithmic equation like \( 2 \log x = 6 \), our goal is to determine the value of \( x \) that satisfies this equation. Solving for \( x \) involves isolating \( x \) on one side of the equation through a series of logical mathematical steps.

• Isolate the Logarithm: The first step is to isolate the logarithmic expression. In our equation, \( 2 \log x = 6 \), we can divide both sides by 2 to simplify: \[ \log x = \frac{6}{2} = 3 \]
• Convert to Exponential Form: With the logarithm isolated, rewrite the equation in exponential form (as detailed below) to find \( x \).

Approaching equations step-by-step ensures clarity and makes it easier to handle similar problems in the future.

Exponential Form

Once the logarithm \( \log x = 3 \) is isolated, it's time to convert this logarithmic equation into its exponential form. Doing so provides a straightforward way to evaluate \( x \).The key idea here is to use the definition of logarithms: If \( \log x = y \), then \( x = 10^y \) when dealing with common logarithms, which have a base of 10.

• Convert the Logarithm: For our equation, \( \log x = 3 \), we rewrite this as \( x = 10^3 \). This transformation uses the base of the logarithm (10 in this case) to express \( x \) as a power.
• Evaluate the Expression: Calculate the power to find the value of \( x \): \[ x = 1000 \]

Converting to exponential form simplifies solving because it directly leads to the value of \( x \) without needing further logarithmic manipulations.

Common Logarithm

In our problem, the presence of a common logarithm simplifies the transition between logarithmic and exponential forms. A common logarithm has a base of 10, which is why it is often written simply as \( \log \) rather than \( \log_{10} \).

• Understanding Base 10: The common logarithm is specifically referenced when numbers are powers of 10. This feature is what allows equations like \( \log x = 3 \) to be quickly converted into \( x = 10^3 \).
• Simplification: Using a common logarithm can simplify calculations, especially in contexts where calculations or conversions frequently involve powers of 10.

Knowing the characteristics of common logarithms makes it easier to tackle equations involving logarithms, reinforcing the connection between logarithms and exponential expressions.