Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to solve the equation. $$\log _{4}(5 x-4)=2$$

Step-by-Step Solution

Answer

We can solve the equation \(\log_{4}(5x - 4) = 2\) using the laws of logarithms: 1. Expanding the equation using the definition of logarithm, we get \(4^2 = 5x - 4\). 2. Simplifying the equation we get, \(16 = 5x - 4\). 3. Solving for x, we find that \(x = 4\).

Step 1: Use the definition of logarithm to rewrite the equation

Recall that a logarithm is the inverse of an exponent. We can rewrite the logarithmic equation in exponential form using the definition of a logarithm: if \(\log_{a}(b) = c\), then \(a^c = b\). Applying this to our given equation, we get: \(4^2 = 5x - 4\)

Step 2: Simplify the exponential form of the equation

Now we can simplify the equation: \(4^2 = 16\) So, our equation becomes: \(16 = 5x - 4\)

Step 3: Solve for x

To solve for x, we need to isolate it. First, let's add 4 to both sides of the equation: \(16 + 4 = 5x\) \(20 = 5x\) Now, we can divide both sides by 5 to find the value of x: \(x = \frac{20}{5}\) \(x = 4\) So, the solution to the given logarithmic equation is \(x = 4\).

Key Concepts

Exponential Form

Converting a logarithmic equation into its exponential form is a crucial step in solving these types of problems. This step uses the definition of a logarithm, which relates the two forms. Logarithms and exponents are inverses, meaning:

• If you have \(\log_{a}(b) = c\), it equates to \(a^c = b\).

In the context of our original exercise, \(\log_{4}(5x - 4) = 2\), we translate it into an exponential form to make the problem more approachable. By using the definition, we rewrite it as:\[4^2 = 5x - 4\]This transition transforms a potentially complex logarithmic expression into a simpler algebraic equation. Recognizing this step allows us to directly proceed by solving for the unknown variable after performing basic arithmetic calculations.

Laws of Logarithms

The laws of logarithms are essential tools that often simplify logarithmic expressions and equations. Although not extensively used in our exercise, understanding these rules can greatly aid in solving various logarithmic problems.

Here are some of the primary laws:

• **Product Rule:** \(\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)\)
• **Quotient Rule:** \(\log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c)\)
• **Power Rule:** \(\log_{a}(b^c) = c \cdot \log_{a}(b)\)

These properties help simplify complex logarithmic expressions into simpler forms, which can then be converted into exponential forms as needed. Mastering these laws makes tackling logarithmic equations a more straightforward task.

Solving Equations

Once you've converted a logarithmic equation to its exponential form, the next step is solving for the variable through algebraic manipulation. In our example, after translating \(\log_{4}(5x - 4) = 2\), we obtained the exponential form:\[4^2 = 5x - 4\]To solve for \(x\), follow these steps:

• Calculate \(4^2\), giving 16, which simplifies the equation to \(16 = 5x - 4\).
• Add 4 to both sides to get \(20 = 5x\).
• Finally, divide both sides by 5 to isolate \(x\): \(x = \frac{20}{5} = 4\).

Each step creatively uses basic arithmetic to gradually zero in on the solution. Understanding each manipulation ensures you can handle more elaborate equations in logarithmic contexts.