Problem 81 - Algebra and Trigonometry

Question

Solve each logarithmic equation. 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. $$ 2 \log x=\log 25 $$

Step-by-Step Solution

Answer

The solution to the logarithmic equation is \(x = 12.5\).

Step 1: Simplify the Equation

From the given equation \(2 \log x = \log 25\), divide through by 2 to get \(\log x = \log 25/2\). This simplifies further to \(\log x = \log 12.5\).

Step 2: Use Logarithm Properties

If \(\log a = \log b\), then \(a = b\). By applying this logarithm property, \(\log x = \log 12.5\) becomes \(x = 12.5\).

Step 3: Check the Domain

In logarithmic expressions, \(x > 0\). Therefore, for the original expression, \(x = 12.5\) is a valid solution because it fits within the domain.

Step 4: Decimal Approximation

The exact answer \(x = 12.5\) is already rounded to the nearest two decimal places, so no further approximation is necessary.

Key Concepts

Solving Logarithmic Equations

In order to tackle logarithmic equations, it's essential first to understand what a logarithm is. A logarithm with a given base answers the question: 'To what power must we raise this base to obtain a certain number?'. For instance, if you're faced with the logarithmic equation \( 2 \log x = \log 25 \), the objective is to find the value of the variable \(x\) that makes the equation true.

Let's break down the solution process for the example provided. Initially, the equation can be simplified by using the property that allows us to divide each term by the same non-zero number. In this case, dividing by 2 gives us \( \log x = \log 12.5 \). Following the simplification, the logarithms on both sides of the equation allow us to compare their arguments directly because of the property that states if \( \log a = \log b \), then \(a = b\). This means that for \(x\) to satisfy the original equation, it must be equal to 12.5.

When dealing with logarithms, remember always to check that the solution falls within the acceptable range or domain for logarithms, which is all positive real numbers. This process ensures that the value of \(x\) you've found is valid.

Logarithm Properties

Logarithms boast several unique properties that are exceptionally helpful when solving logarithmic equations. One such property is the 'one-to-one' property, which states that if \( \log_b(a) = \log_b(c) \), then \(a = c\). This was utilized in our exercise to arrive at the solution \(x = 12.5\).

Other important properties include:

• Product Property: \( \log_b(a \cdot c) = \log_b(a) + \log_b(c) \)
• Quotient Property: \( \log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c) \)
• Power Property: \( \log_b(a^c) = c \cdot \log_b(a) \)

These properties are instrumental in simplifying complex logarithmic expressions and solving for variables within them. In the context of our original problem, we could have also used the Power Property in the first step before dividing by 2, which would yield the same result.

Domain of Logarithmic Functions

The domain of a logarithmic function is crucial when solving logarithmic equations. A logarithm is only defined for positive real numbers. This is because you cannot take a log of a negative number or zero in the real number system. The base of a logarithm also has important restrictions; it must be a positive real number other than 1.

In the exercise at hand, the domain consideration comes into play when verifying our solution. Since the argument of the logarithm is \(x\), and logarithms are only valid for \(x > 0\), any solution we find must be a positive number. Fortunately, the solution we derived, \(x = 12.5\), is indeed a positive number, which means it falls within the domain of the logarithmic function and is therefore valid.