Problem 42 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(6 .\) \(\log 5 x=4\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2000\).

Step 1: Identify the Base of the Logarithm

In the equation \(\log 5x = 4\), the logarithm is implied to be base 10. This means the equation can be interpreted as \(\log_{10}(5x) = 4\).

Step 2: Convert the Logarithmic Equation to Exponential Form

Using the property of logarithms, \(\log_b(a) = c\) implies \(b^c = a\), we convert \(\log_{10}(5x) = 4\) into its exponential form: \(10^4 = 5x\).

Step 3: Solve for x

Calculate \(10^4\), which equals 10000. Then, use this to solve for \(x\) in the equation:\[5x = 10000\] Divide both sides by 5 to isolate \(x\):\[x = \frac{10000}{5} \].Simplify to find \(x = 2000\).

Key Concepts

Understanding the Base of Logarithm

When working with logarithmic equations, such as \( \log 5x = 4 \), it's crucial to first identify the base of the logarithm. In most typical logarithms found in basic math problems, the base is implied to be 10, unless otherwise stated. This is known as the "common logarithm."

• So, when you see an equation like \( \log 5x = 4 \), you can interpret it as \( \log_{10}(5x) = 4 \).
• The base of 10 is widely used in various scientific and engineering fields due to its alignment with the decimal system.

Having a solid understanding of what the base represents helps you significantly when converting logarithmic equations into exponential form— a key step in solving these types of problems. Remember, if the base is not explicitly mentioned, it's safe to assume it is 10.

Converting to Exponential Form

One of the most powerful tools in solving logarithmic equations is the ability to convert them into exponential form. This method makes equations easier to solve. For example, say we're working with the equation \( \log_{10}(5x) = 4 \).

• The rule to remember is: if \( \log_b(a) = c \), then it can be rewritten as \( b^c = a \).
• This translates our equation into \( 10^4 = 5x \).

Switching to exponential form means you're entwining your understanding of both logarithms and exponents into one coherent method. You take a seemingly complex logarithmic question and boil it down to a simpler arithmetic problem, thereby making the equation more approachable.Understanding how and when to use this conversion is a vital skill in solving logarithmic equations, and brings clarity to problems that may initially seem complex.

Solving the Equation

After converting the logarithmic equation into exponential form, the next step is straightforward: solve the equation like any other algebra problem. In our example, we reached the equation \( 10^4 = 5x \) which simplifies to \( 10,000 = 5x \).Here's how you tackle it:

• Calculate \( 10^4 \), which equals 10,000.
• Set up the equation \( 5x = 10,000 \).
• To solve for \( x \), divide both sides by 5: \( x = \frac{10,000}{5} \).

This division gives us \( x = 2000 \).The process involves basic arithmetic: calculating powers and performing division. With practice, this flow becomes intuitive, allowing you to handle increasingly complex logarithmic expressions. The important part is to systematically approach each problem in this ordered manner, ensuring you don’t miss critical steps along the way.