Problem 77 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log (x+4)=\log x+\log 4 $$

Step-by-Step Solution

Answer

\(x = 1.33\)

Step 1: Simplify the equation

Use the property of logarithms that states \(\log_a{mn}=\log_a{m}+\log_a{n}\). Given \(\log (x+4)=\log x+\log 4\), we translate the right side of equation into singular logarithmic form: \( \log (4x) \). Therefore, our updated equation is \(\log (x+4)=\log (4x)\).

Step 2: Remove the logarithms

Once we have the same logarithm on both sides, we can eliminate them. This gives a new equation to solve: \(x + 4 = 4x\).

Step 3: Solve for \(x\)

Subtract \(x\) from both sides of the equation to find \(4 = 3x\), and then divide each side by 3, which gives \(x = 4/3\).

Step 4: Check if the solution is within the domain

The original equation was \(\log (x+4)=\log x+\log 4\). The domain of a logarithmic function is greater than 0. Substituting \(x = 4/3\) shows that this value is a valid solution for both logarithmic expressions, since \(4/3 + 4 > 0\) and \(4/3 > 0\).

Step 5: Decimal approximation

Although not strictly necessary in this case as \(x = 4/3\) is the exact answer, the question also ask for a decimal approximations up to two decimal places. So, \(x = 4/3 = 1.33\) when rounded off to two decimal places.

Key Concepts

Properties of Logarithms

When dealing with logarithmic equations, understanding the **properties of logarithms** is key to simplifying and solving them. One crucial property is that the logarithm of a product can be separated into the sum of two logarithms: \( \log_a{mn} = \log_a{m} + \log_a{n} \). This property allows us to combine or split logarithms, making equations more manageable.
For example, in the equation \( \log(x+4)=\log x+\log 4 \), we use this property to combine the right side into \( \log(4x) \). Some other useful properties to remember include:

• \( \log_a{\frac{m}{n}} = \log_a{m} - \log_a{n} \) (Quotient Rule)
• \( \log_a{m^n} = n \cdot \log_a{m} \) (Power Rule)

Applying these properties can drastically simplify complex logarithmic expressions, leading you towards the solution.

Solving Equations

Once logarithmic expressions are simplified, the next step is **solving the equation**. If both sides of the equation have the same logarithm, it can often be removed, making solving straightforward. This occurs because a logarithmic function is one-to-one, meaning if \( \log(m) = \log(n) \), then \( m = n \).
In our exercise, after rewriting the equation as \( \log(x+4)=\log(4x) \), we remove the logarithms resulting in a simpler linear equation: \( x+4=4x \). From here, it’s all about basic algebra:

• Subtract \( x \) from both sides: \( 4 = 3x \)
• Divide each side by 3: \( x = \frac{4}{3} \)

These steps help finalize the solution, but it's important to remember to check if this solution fits within the mathematical constraints of the problem.

Mathematical Domain

**Mathematical domain** is a critical factor, especially in logarithmic equations, as it defines the set of values for which the equation is valid. Logarithms are only defined for positive numbers, meaning any argument of a logarithmic function must be greater than zero. It's crucial to verify that potential solutions do not result in taking the logarithm of zero or a negative number.
For the given problem, check both \( \log(x) \) and \( \log(x+4) \):

• \( x > 0 \)
• \( x+4 > 0 \) implies \( x > -4 \)

These checks affirm that the solution \( x = \frac{4}{3} \), which is positive and greater than both boundaries, thus lies within the domain of the original equation.

Decimal Approximation

After solving the equation accurately, you might sometimes need a **decimal approximation** for practical applications or to meet problem requirements. This involves converting a fraction or irrational number into its decimal form.
In our exercise, the exact solution is \( x = \frac{4}{3} \). Nevertheless, the instruction requests a decimal approximation accurate to two decimal places.
To convert \( \frac{4}{3} \) to a decimal:

• Carry out the division: \( 4 \div 3 = 1.3333\ldots \)
• Round to two decimal places: \( 1.33 \)

This approximation gives a simple, easily readable form of the solution where needed, although using exact values retains mathematical precision.