Problem 26 - Technical Mathematics with Calculus

Question

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log x+\log 4 x=2$$

Step-by-Step Solution

Answer

The value of \(x\) is 5.

Step 1: Combine Logarithms

Use the logarithm property \(\log a + \log b = \log(ab)\) to combine the logarithmic expressions into one. So, \(\log x + \log (4x) = \log(4x^2)\).

Step 2: Set Up the Equation

Set the combined logarithm equal to 2: \(\log(4x^2) = 2\).

Step 3: Rewrite in Exponential Form

Rewrite the equation \(\log(4x^2) = 2\) in exponential form to solve for \(x\). The exponential form is \(10^2 = 4x^2\), since the base of the logarithm is 10.

Step 4: Solve for x^2

Solve the equation for \(x^2\) by dividing both sides by 4, giving \(x^2 = 10^2 / 4\) or \(x^2 = 100 / 4\).

Step 5: Take the Square Root

Take the square root of both sides to solve for \(x\). Doing this, we get \(x = \sqrt{25}\) or \(x = -\sqrt{25}\). However, since \(x\) is in a logarithmic expression, it must be positive. Therefore, \(x = \sqrt{25}\).

Step 6: Calculate and Check the Answer

Calculate the square root of 25 to find \(x = 5\). Check the answer by substituting back into the original equation: \(\log(5) + \log(4 \cdot 5)\). After checking, both sides equal 2, confirming the solution.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is fundamental when working with logarithmic equations. A logarithm is an exponent to which a base number must be raised to yield a given number. There are key properties used to manipulate and solve these equations.

Product Property

The product property, which states that \(\log_a(mn) = \log_a(m) + \log_a(n)\), is used to combine two logarithms with the same base that are added together. This property is particularly useful for simplifying logarithmic expressions before solving for variables.

Quotient Property

The quotient property, which states that \(\log_a(\frac{m}{n}) = \log_a(m) - \log_a(n)\), is useful for breaking down a logarithm of a division into a difference of two logarithms.

Power Property

The power property, \(\log_a(m^n) = n\log_a(m)\), allows you to take an exponent within the log and multiply it by the log. This is helpful to isolate the variable when it is an exponent inside the logarithm.

These properties are essential for combining, expanding, or reducing logarithmic terms to a form where the variable of interest can be isolated and solved.

Exponential Form

Transitioning between logarithmic form and exponential form is a key technique in solving logarithmic equations. The exponential form of a logarithm is based on the definition that if \(\log_b(a) = c\), then \(b^c = a\). This allows you to convert the log equation, which may not be straightforward to solve, into an exponential equation, where solving for the variable often becomes simpler.

For example, if we have the equation \(\log(4x^2) = 2\), we know that the base of the logarithm is 10 (since it's not specified). We can thus rewrite this equation in exponential form as \(10^2 = 4x^2\). This approach often paves the way to a solution by allowing basic algebraic operations to be used to isolate and solve for the variable.

Solving for x

Solving for 'x' in equations, especially in the context of logarithms, often requires a few changes in form and strategic use of properties. After using properties of logarithms to combine or simplify terms, you usually rewrite the logarithmic equation in exponential form. This can lead to a basic algebraic expression where 'x' may appear as a factor or within an exponent.

In the example \(10^2 = 4x^2\), solving for \(x^2\) involves straightforward algebra—dividing both sides by 4. Once \(x^2\) is isolated, obtaining 'x' requires taking the square root. It's crucial to remember that when dealing with logarithms, only the positive square root is valid, as logs are not defined for negative numbers or zero.

To ensure accuracy, always substitute the solved value(s) back into the original equation to check if they truly satisfy the equation. This step confirms whether the solution is correct and can reveal any extraneous roots, which are apparent solutions that do not actually satisfy the original equation.