Problem 41 - Intermediate Algebra

Question

Solve each equation. $$ \log 2 x=4 $$

Step-by-Step Solution

Answer

The solution to \( \log 2x = 4 \) is \( x = 5000 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log (2x) = 4 \). Here, we have the logarithm (base 10 is implied when no base is mentioned in the log). The task is to solve for \( x \).

Step 2: Convert the Logarithmic Equation to an Exponential Equation

The equation \( \log_{10} (2x) = 4 \) can be rewritten as an exponential equation: \( 2x = 10^4 \). This conversion is done because if \( \log_{b}(A) = C \), then \( A = b^C \).

Step 3: Calculate the Power of 10

Calculate \( 10^4 \):\[ 10^4 = 10000 \] Thus, the equation becomes \( 2x = 10000 \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation \( 2x = 10000 \) by 2: \[ x = \frac{10000}{2} \] \[ x = 5000 \]

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent. In solving such equations, the goal is to isolate the variable. These types of equations look like this: \( b^x = A \), where \( b \) is the base and \( A \) is a constant. In the example problem, the base is 10, and the equation simplifies to \( 2x = 10^4 \). You'll usually need to manipulate these equations to either compare like bases or use logarithms to find the exact value of the variable. When dealing with base 10, it can be straightforward because you can convert back and forth between logarithmic and exponential forms. Understanding how to rewrite the equation is crucial in making the problem simpler.

Solving Equations

To solve equations means finding the value of the unknown variable that makes the equation true. Various methods are employed depending on the type of equation. In the context of logarithmic or exponential equations, rewriting the equation into a more familiar form is often part of the solution process. For the logarithmic equation \( \log (2x) = 4 \), converting it into the exponential form \( 2x = 10^4 \) is a key step. This allows us to solve for \( x \) directly. After conversion, solving \( 2x = 10000 \) is straightforward: merely divide both sides by 2. As a final result, \( x = 5000 \). Remember:

• Convert the equation into a simpler form.
• Isolate the variable.
• Solve the equation using arithmetic principles.

Breaking down steps like this makes it easier to handle complex mathematical operations efficiently.

Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, are logarithms with a base of 10. Often in mathematics, when you see \( \log \) written without assuming a base, it defaults to 10. This is particularly useful for scales like Richter, pH, and powers of 10 in scientific notation. When dealing with base 10 logs, the conversion to an exponential form is an essential technique. This is because the property \( \log_{10} (A) = C \) is equivalent to \( A = 10^C \). Using this rule, \( \log (2x) = 4 \) translates to \( 2x = 10^4 \). Understanding how to move between logarithmic and exponential forms is advantageous in simplifying and solving equations efficiently. Base 10 is easy to work with manually, due to its decimal nature—making such operations intuitive once you grasp the basic conversions.