Problem 37 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x\) $$ \log x=-2 $$

Step-by-Step Solution

Answer

x = 0.01

Step 1: Understand the logarithmic equation

The given equation is \(\log x = -2\). This equation is in base 10, meaning it is shorthand for \(\log_{10} x = -2\). We need to find the value of \(x\) that satisfies this equation.

Step 2: Convert the logarithmic equation to exponential form

To solve the logarithmic equation, we convert it into its equivalent exponential form. The equation \(\log_{10} x = -2\) translates to \(x = 10^{-2}\). This is because if \(\log_{10} x = y\), then \(x = 10^y\).

Step 3: Calculate the value of \(x\)

Now compute the value of \(x = 10^{-2}\). This means we take the reciprocal of 10 squared, which is \(\frac{1}{10^2}\), resulting in \(x = \frac{1}{100} = 0.01\).

Step 4: Verify the solution

We can verify our solution by plugging \(x = 0.01\) back into the original equation. Calculate \(\log_{10} 0.01\). Since \(10^{-2} = 0.01\), the calculation is correct as \(\log_{10} (10^{-2}) = -2\). Therefore, \(x = 0.01\) is indeed the solution.

Key Concepts

Exponential Form

When you encounter a logarithmic equation like \( \log_{10} x = -2 \), it's useful to convert it into what we call "exponential form." This transformation is fundamental because it helps you to directly solve for the unknown variable, in this case, \( x \). The key idea is to rewrite the logarithmic equation using the base of the logarithm as the base of the exponent in the new equation.
In general, if you have \( \log_b x = y \), the equivalent exponential form is \( x = b^y \). For our specific problem, the base \( b \) is 10, so our equation \( \log_{10} x = -2 \) becomes \( x = 10^{-2} \). Now you can clearly see that solving for \( x \) involves calculating the value of \( 10^{-2} \), which simplifies the task significantly.

Base 10 Logarithms

Base 10 logarithms, often written simply as \( \log \), are logarithms that use 10 as their base. They are particularly popular in fields like science and engineering because they correspond to the decimal numbering system.
Let's delve a bit deeper into the concept. The expression \( \log_{10} x \) answers the question: "To what power must 10 be raised, to yield \( x \)?" It's a way of translating between multiplication and repeated addition. In our exercise, the equation \( \log_{10} x = -2 \) is asking to find \( x \) such that raising 10 to the power of \(-2\) equals \( x \).

• Negative Exponent: A negative exponent means you're actually dealing with a fraction. Specifically, \( 10^{-2} \) means \( \frac{1}{10^2} \).
• Key Feature: Because it involves the base 10, it's also called the common logarithm, widely used without needing to specify the base.

Solution Verification

Verifying your solution to a mathematical problem is a critical step. It ensures that the result obtained is indeed correct and satisfies the original equation.
After finding \( x = 0.01 \) as a solution, you should substitute it back into the original logarithmic equation to check if it holds true. In this case, substitute \( x = 0.01 \) into \( \log_{10} x = -2 \).
To verify, calculate \( \log_{10}(0.01) \). Knowing \( 0.01 \) is equivalent to \( 10^{-2} \), we can recognize that \( \log_{10}(10^{-2}) \) equals -2, which is indeed our original equation. Thus, this confirms that \( x = 0.01 \) satisfies the entire equation.

• Accuracy: Having this verification step enhances the reliability of your solution.
• Clarity: Reassures that you understand not just how to solve the problem but also why the solution works.