Problem 40 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x.\) \(\log (x-4)=3\)

Step-by-Step Solution

Answer

The solution is \(x = 1004\).

Step 1: Understand the Logarithmic Equation

The given equation is \(\log (x-4) = 3\). This is a standard logarithmic equation where our goal is to solve for \(x\).

Step 2: Change Logarithmic Equation to Exponential Form

Recall that the equation \(\log_b(a) = c\) can be rewritten in exponential form as \(b^c = a\). Since no base is provided, we assume base 10, which gives \(10^3 = x-4\).

Step 3: Calculate the Exponential Value

Compute \(10^3\). This equals 1000. Therefore, the equation now is \(x - 4 = 1000\).

Step 4: Solve for \(x\)

To isolate \(x\), add 4 to both sides of the equation: \(x = 1000 + 4\).

Step 5: Simplify the Expression

Calculate \(1000 + 4\) to get \(x = 1004\).

Key Concepts

Solving Logarithmic Equations

Logarithmic equations can seem daunting at first, but they're all about understanding the relationship between logs and exponentials. The goal in solving a logarithmic equation is typically to find the value of the variable that makes the equation true. We start by analyzing the given logarithmic equation. For example, in the equation \( \log (x-4) = 3 \), our task is to isolate the variable \( x \). To do this, we often need to convert the logarithmic equation into its equivalent exponential form.

• Check if the logarithmic equation is in a standard form such as \( \log_b(a) = c \).
• If the base isn't specified, assume it to be 10, which is the common logarithm base.
• Convert the equation into exponential form to make it easier to solve for the variable.

By following these steps, you can reduce the complexity of the equation, allowing you to zero in on the value of the unknown variable.

Exponential Form

Converting a logarithmic equation into exponential form is a crucial step when solving the equation. The logarithmic form \( \log_b(a) = c \) expresses the idea that \( b \) raised to the power of \( c \) gives \( a \). In our example, we had \( \log (x-4) = 3 \). We assume a base of 10 here.

• Rewriting the equation, we get \( 10^3 = x - 4 \). This transformation is key because it shifts our perspective from dealing with logs to dealing with powers.
• Once in exponential form, it becomes simpler to solve for the unknown since you're often performing basic arithmetic operations.
• After converting, calculate the power to determine the value of \( a \).

By converting to exponential form, logarithmic equations become more manageable, making it easier to solve for the variable.

Base 10 Logarithms

The base 10 logarithm, often referred to as the common logarithm, is prevalent in many mathematical applications. It provides a standard way to deal with power functions and simplifies calculations.

• The base 10 logarithm is implied whenever you see \( \log \) without a base indicated, like in \( \log (x-4) = 3 \). This means we're working with powers of 10.
• This type of log is useful because we frequently work with the decimal system, and powers of 10 are fundamental here.
• Using base 10 makes mental calculations and estimations easier.

Understanding the context of base 10 logarithms allows us to interpret and solve logarithmic equations more effectively, as demonstrated in our example, where we assumed base 10 to solve \( \log (x-4) = 3 \).