Problem 41 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(6 .\) $$ \log 2 x=4 $$

Step-by-Step Solution

Answer

The solution is \( x = 5000 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log 2x = 4 \). This means that the logarithm of some number \( 2x \) is equal to 4, where the base of the logarithm, if not specified, is assumed to be 10.

Step 2: Convert the Logarithmic Equation to Exponential Form

According to the definition of logarithms, \( \log_b a = c \) is equivalent to \( a = b^c \). In this problem, \( \log 2x = 4 \) implies \( 2x = 10^4 \), since the base \( b \) is 10.

Step 3: Simplify the Exponential Equation

Calculate \( 10^4 \). This results in \( 10^4 = 10000 \). Thus, the equation becomes \( 2x = 10000 \).

Step 4: Solve for \( x \)

To isolate \( x \), divide both sides of the equation by 2. This gives \( x = \frac{10000}{2} \).

Step 5: Calculate the Final Value of \( x \)

Perform the division: \( \frac{10000}{2} = 5000 \). Therefore, \(x = 5000\).

Key Concepts

Exponential Form

To effectively tackle logarithmic equations, it's essential to understand the concept of exponential form. When you see a logarithmic statement like \( \log_b a = c \), it can be transformed into an exponential equation: \( a = b^c \). This conversion is vital because it gives us a direct way to calculate the value we need. In exponential form, we're essentially saying _what number \( a \) equals when raised to the power \( c \) using base \( b \)._

For example, in the equation \( \log 2x = 4 \), the base \( b \) is implied to be 10 (as it's common to use base 10 in logarithms when no base is shown). Therefore, the equation converts to \( 2x = 10^4 \), meaning \( 2x \) equals 10 raised to the power of 4. By converting, we have transformed the logarithmic equation into a more straightforward exponential form, where it's much easier to solve for the unknown variable.

Solving Equations

Solving equations, particularly logarithmic ones, involves a step-by-step approach to isolate the variable you're looking for. Once you've converted the logarithmic equation into its exponential form, the equation becomes simpler to work with.

After representing \( \log 2x = 4 \) as \( 2x = 10^4 \), the next step is to calculate \( 10^4 \). In arithmetic, \( 10^4 \) means multiplying 10 by itself four times, which equals 10,000. This reduces the equation to \( 2x = 10,000 \).

To find \( x \), you must isolate it by performing operations that undo the multiplication. Since \( x \) is multiplied by 2, divide both sides of the equation by 2. This simplifies the equation further to \( x = \frac{10,000}{2} \), resulting in \( x = 5,000 \). By following these logical steps, we solve the equation efficiently, arriving at the desired solution.

Logarithms

Logarithms are powerful mathematical tools related to exponentiation. They answer the question: _"To what exponent must we raise a base, often 10 or \( e \), to get a specific number?"_ The notation \( \log_b a = c \) reads as "\( c \) is the power to which the base \( b \) must be raised to yield \( a \)." This relationship is why logarithms are deeply connected to exponential functions.

When working with logarithms, it's crucial to remember that every logarithm statement can be translated into an exponential statement, thus making computations and conceptual understanding easier. In everyday practical use, logarithms help in handling equations that involve exponential growth or decay, such as calculating compound interest, population growth, or radioactive decay.

In our original problem, \( \log 2x = 4 \), we effectively used the properties of logarithms to switch to an exponential equation. This transformation simplifies the process of solving for \( x \). Understanding logarithms' underlying principles breaks down the seeming complexity into manageable steps, reinforcing why logarithms are considered foundational in math and applied sciences.