Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ \log x^{2}=-4 $$

Step-by-Step Solution

Answer

The solution is \( x = 0.01 \).

Step 1: Understand the Equation

The equation given is \( \log x^{2} = -4 \). Here, \( \log \) refers to the common logarithm, which is logarithm with base 10. Our task is to find the value of \( x \) that satisfies this equation.

Step 2: Use Logarithm Properties

Recall that \( \log a^b = b \log a \). We can rewrite the given equation \( \log x^{2} = -4 \) as \( 2 \log x = -4 \).

Step 3: Isolate \( \log x \)

Divide both sides of the equation \( 2 \log x = -4 \) by 2 to isolate \( \log x \). We get \( \log x = -2 \).

Step 4: Convert to Exponential Form

To solve for \( x \), convert the logarithmic equation \( \log x = -2 \) to its exponential form using the definition of a logarithm: if \( \log_{b} y = a \), then \( y = b^a \). So \( x = 10^{-2} \).

Step 5: Calculate the Value of \( x \)

Calculate \( 10^{-2} \), which is the same as \( \frac{1}{10^2} \), resulting in \( x = 0.01 \).

Key Concepts

Common Logarithm

When we talk about common logarithms, we're dealing with logarithms that have a base of 10. The notation used for common logarithms is generally \( \log \), without a base indicated, because base 10 is assumed. Common logarithms are frequently used in sciences and engineering because they relate to the decimal number system that we are so familiar with. This makes them convenient when dealing with powers of 10. In the exercise provided, the equation \( \log x^{2} = -4 \) uses a common logarithm to express that the power to which 10 must be raised to yield \( x^2 \) is \(-4\). Understanding that \( \log \) here means base 10 is crucial because it informs the process we use to solve for \( x \).

Exponential Form

Converting a logarithmic equation into its exponential form can simplify the process of solving the equation. When you see an equation like \( \log x = -2 \), you're actually seeing a relationship that can be expressed exponentially.To convert this equation into its exponential form, use the definition of a logarithm: if \( \log_{b} y = a \), then it implies \( y = b^a \). In the common logarithm with base 10, \( \log_{10} x = -2 \) becomes \( x = 10^{-2} \). This exponential transformation is often helpful because it allows us to see the number \( x \) in terms of a straightforward power of 10, which is usually easier to compute and understand.

Properties of Logarithms

Understanding properties of logarithms allows one to manipulate and solve logarithmic equations more effectively.One crucial property is the power rule for logarithms: \( \log a^b = b \log a \). This means that when you have a term like \( \log x^2 \), you can rewrite it as \( 2 \log x \). It helps simplify computations and convert complex expressions into manageable forms. Another key property is that of equality: if \( \log a = \log b \), then it must be that \( a = b \). This property is used to isolate the variable and solve the equation. In the problem \( \log x^{2} = -4 \), using the power rule allows us to express the equation as \( 2 \log x = -4 \), simplifying our work in finding \( x \). By dividing by 2, we isolate \( \log x \), eventually solving for \( x \) efficiently by converting to an exponential form.