Problem 36 - Algebra 2

Question

Solve each equation. Check your answers. $$ \log x+4=8 $$

Step-by-Step Solution

Answer

The solution of the equation is \( x = 10,000 \).

Step 1: Isolate the logarithm

First, we must isolate the logarithm on one side of the equation. We can do this by subtracting 4 from each side of the equation. This will result in the new equation: \( \log x = 8 - 4 \) or \( \log x = 4 \).

Step 2: Convert to exponential form

Now, convert the equation from log to exponential form to remove the log and simplify the problem. The resulting equation is \( x = 10^4 \).

Step 3: Solve for x

To solve for x, we simply calculate the power \( 10^4 \), which gives us \( x = 10,000 \).

Step 4: Check the solution

The last step is to ensure that the solution is correct by substituting it back into the original equation. This allows us to see if the equation holds for \( x = 10,000 \). When we insert \( x = 10,000 \) into the initial equation it results in \( \log 10,000 + 4 = 8 \). Since \( \log 10,000 = 4 \), we get \( 4 + 4 = 8 \). It is true, so \( x = 10,000 \) is indeed the correct solution.

Key Concepts

Isolating the Logarithm

When solving logarithmic equations, one of the primary objectives is to isolate the logarithm. This means you want the log term to be by itself on one side of the equation, just like you would isolate a variable in algebra to solve for it. In our example, the equation is \( \log x + 4 = 8 \). To isolate \( \log x \), we subtract 4 from both sides. This step is straightforward and involves basic arithmetic:

• Start with the original equation: \( \log x + 4 = 8 \)
• Subtract 4 from each side: \( \log x = 8 - 4 \)
• Simplify to find \( \log x = 4 \)

Once the logarithm is isolated, you are ready to convert the equation into another form, which makes it easier to solve.

Exponential Form Conversion

Converting the logarithmic equation into exponential form is crucial for solving equations involving logarithms. This process involves interpreting the equation \( \log x = 4 \) into a form that allows you to solve for \( x \) directly. The logarithmic expression \( \log x = 4 \) is based on the common logarithm, which uses base 10. This means

• The logarithm states that \( x \) is the number 10 must be raised to in order to result in 4.
• In exponential form, this becomes \( x = 10^4 \).
• Calculating \( 10^4 \), you find that \( x = 10,000 \).

Converting to exponential form turns a complex logarithmic equation into a simpler arithmetic problem, showing the power of transformation in solving these equations.

Checking Solutions

After you have calculated the solution for \( x \), it's important to verify if the solution is correct. This helps to confirm that your steps and calculations were accurate. In our example, once you have determined that \( x = 10,000 \), insert this back into the original equation:

• Original equation: \( \log x + 4 = 8 \)
• Substitute \( x = 10,000 \): \( \log 10,000 + 4 \)
• Find \( \log 10,000 \): since \( 10,000 = 10^4 \), \( \log 10,000 = 4 \)
• Add 4 to check: \( 4 + 4 = 8 \)

The equation holds true, confirming that our computed solution for \( x \) is indeed correct. This step of checking solutions ensures accuracy and builds confidence in your ability to solve exponential and logarithmic equations.