Problem 19 - Technical Mathematics with Calculus

Question

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log (2 x+5)=2$$

Step-by-Step Solution

Answer

The value of \(x\) to three significant digits is 47.5.

Step 1: Apply the Definition of the Logarithm

The equation \(\log (2x + 5) = 2\) can be rewritten in exponential form as \(10^2 = 2x + 5\). This uses the definition of logarithms, which states that \(\log_b a = c\) implies \(b^c = a\). Here, the base \(b\) is 10 because we have a common logarithm.

Step 2: Simplify the Exponential Expression

Calculate \(10^2\) to simplify the right-hand side of the equation. Thus, the equation becomes \(100 = 2x + 5\).

Step 3: Isolate the Variable

To solve for \(x\), subtract 5 from both sides of the equation to isolate terms with \(x\) on one side: \(100 - 5 = 2x\), which simplifies to \(95 = 2x\).

Step 4: Solve for \(x\)

Divide both sides of the equation by 2 to solve for \(x\): \(\frac{95}{2} = x\), which simplifies to \(x = 47.5\).

Step 5: Check the Solution

Substitute \(x = 47.5\) back into the original equation to check the solution: \(\log (2(47.5) + 5) = \log (95 + 5) = \log (100)\). Since \(\log (100) = 2\), the check confirms that our solution is correct.

Key Concepts

Logarithm Definition

Understanding logarithms is essential when solving equations like \(\log (2x + 5) = 2\). A logarithm answers the question: to what exponent must we raise a given base to produce a certain number? Defined mathematically, if \(\log_b a = c\), this means the base \(b\) raised to the power \(c\) will give \(a\). In the expression \(\log (2x + 5)\), the base is assumed to be 10, since no base is written. This is called a 'common logarithm'.

Why is this important? Any time you encounter a logarithm in an equation, you're actually dealing with an exponential relationship in disguise, which can often be solved by rewriting it in exponential form. This is especially handy when the logarithm is the only operation on one side of the equation.

Exponential Form

Transitioning from logarithmic to exponential form is a staple when handling equations involving logarithms. For our example \(\log (2x + 5) = 2\), we express this as \(10^2 = 2x + 5\). This conversion is based on the relationship that \(\log_b a = c\) is equivalent to \(b^c = a\).

In simpler terms, exponential form involves two key parts: the base and the exponent. In the common logarithm case, the base is 10, and we're searching for the power to which this base must be raised in order to yield our target value.

Application to Our Exercise

For our example, moving to the exponential form simplifies the process of solving for \(x\). By knowing that \(10^2\) equals 100, we directly relate the logarithmic equation to a basic linear equation, making our steps towards finding \(x\) much clearer and manageable.

Isolating Variables

The core of solving equations is often about isolating the unknown variable. This means rearranging the equation to get the unknown on one side by itself. In the solution process for \(\log (2x + 5) = 2\), we first converted to exponential form and simplified, getting \(100 = 2x + 5\). The next phase is to isolate \(x\).

We achieve this by performing inverse operations that 'undo' the operations applied to \(x\). Since \(x\) is multiplied by 2 and increased by 5, we reverse these steps: subtracting 5 from both sides, and then dividing by 2. Now, \(x\) stands alone as \(x = \frac{95}{2}\) or \(x = 47.5\).

Why Isolating Variables Matters

Isolating the variable is a method that applies to all types of equations, not only to those involving logarithms. Mastery of this technique is crucial since it's often the final series of steps in finding the solution to an equation. By systematically reversing operations, we move from complex relationships to simple, actionable solutions.