Problem 25 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \log _{10} \frac{1}{x^{2}}=2 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1}{10} \) and \( x = -\frac{1}{10} \).

Step 1: Understand the Equation

We are given the equation \( \log_{10} \frac{1}{x^2} = 2 \). This means we are looking for the value of \( x \) such that the base 10 logarithm of the fraction \( \frac{1}{x^2} \) equals 2.

Step 2: Convert Logarithmic Form to Exponential Form

The equation \( \log_{10} \frac{1}{x^2} = 2 \) can be rewritten in its exponential form. This gives us \( \frac{1}{x^2} = 10^2 \), which simplifies to \( \frac{1}{x^2} = 100 \).

Step 3: Solve for \( x^2 \)

To isolate \( x^2 \), take the reciprocal of both sides of the equation from Step 2. This gives \( x^2 = \frac{1}{100} \).

Step 4: Solve for \( x \)

To find \( x \), take the square root of both sides of the equation. This gives \( x = \pm \sqrt{\frac{1}{100}} \), which simplifies further to \( x = \pm \frac{1}{10} \).

Key Concepts

Exponential Form

To understand how to manipulate logarithmic equations, converting them into exponential form is crucial. This is an essential step because the exponential form often simplifies the components of the equation, making it easier to solve.When dealing with logarithms, remember the definition: the log base 'b' of 'a' (\( \log_b(a) \)) equals 'c' if and only if 'b' raised to the power of 'c' equals 'a'.\[ b^c = a \]For our original equation \( \log_{10} \left( \frac{1}{x^2} \right) = 2 \), applying this definition means rewriting the equation as \( \frac{1}{x^2} = 10^2 \), which is the exponential form.

• This form states that \( 10 \) raised to the power of \( 2 \) results in \( \frac{1}{x^2} \).
• This transformation is key to simplifying the problem and moving towards finding the value of \( x \).

Solving Logarithms

Once in exponential form, solving logarithms becomes as straightforward as algebraic manipulations. In our exercise, once we converted the equation into \( \frac{1}{x^2} = 100 \), the next step involves isolating the unknown variable.To solve for \( x^2 \), take the reciprocal of each side. Doing so flips the equation, giving us:\[ x^2 = \frac{1}{100} \]

• This step turns the focus on simplifying \( x^2 \).
• Once simplified, it paves the way for determining \( x \).

The final step in solving the logarithm involves finding the square root of each side to solve for \( x \). This results in \( x = \pm \frac{1}{10} \), showcasing that logarithms can have more than one solution due to the square root property.

Base 10 Logarithm

In mathematics, the base 10 logarithm, known as the common logarithm, is a crucial tool for simplifying complex calculations, especially those involving powers of 10.The notation \( \log_{10} x \) signifies the power to which the base (10) must be raised to produce the number 'x'. This concept often simplifies expressions in sciences and engineering.

• Using base 10 makes large number calculations more manageable.
• It is particularly helpful because it aligns well with our decimal numeral system and makes interpretation intuitive for base 10 logarithmic expressions.

In our exercise, recognizing the base 10 aspect allowed us to simplify \( 10^2 \) to \( 100 \), facilitating straightforward algebraic manipulation. Understanding how base 10 logarithms function can aid in various spheres beyond math exams, such as computing, electronics, and logarithmic scales in real-world applications.