Problem 41 - Algebra 2

Question

REVIEW If \(\log _{10} x=-3,\) what is the value of \(x ?\) $$ \begin{array}{ll}{\mathbf{F} \quad x=1000} & {\mathbf{H} x=\sqrt{\frac{1}{100}}} \\ {\mathbf{G} x=\frac{1}{1000}} & {\mathbf{J} \quad x=-\sqrt{\frac{1}{100}}}\end{array} $$

Step-by-Step Solution

Answer

The value of \( x \) is \( \frac{1}{1000} \).

Step 1: Understand the logarithmic equation

The problem gives us the equation \( \log_{10} x = -3 \). This implies that \( x \) is the number which, when 10 is raised to the power of \(-3\), gives \( x \). Therefore, we can rewrite this equation as \( x = 10^{-3} \).

Step 2: Simplify the expression

Next, simplify the expression \( 10^{-3} \). This gives us \( x = \frac{1}{10^3} \), or \( x = \frac{1}{1000} \).

Step 3: Match with the multiple choice answers

Compare \( x = \frac{1}{1000} \) with the given options: \( x = 1000 \), \( x = \frac{1}{1000} \), \( x = \sqrt{\frac{1}{100}} \), \( x = -\sqrt{\frac{1}{100}} \). The correct choice matching our computed value is \( x = \frac{1}{1000} \).

Key Concepts

Logarithms

Logarithms are a way to express exponentiation in reverse. They help us ask, "To what power must we raise a base number to obtain another number?" For instance, in the logarithmic notation \( \log_{10} x = -3 \), it is asking, "To what power must 10 be raised to result in \( x \)?" In this exercise, it tells us 10 raised to \( -3 \) equals \( x \).

Some key points about logarithms are:

• The base, often represented as the small number after "log," is the number that gets raised to a certain power.
• The logarithmic equation tells us the exponent needed to reach a given number.

So, when you see \( \log_{10} x = -3 \), it fundamentally translates to \( 10^{-3} = x \). Logarithms can simplify and solve problems that involve exponential relationships by helping us isolate the exponent in expressions.

Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. They are a compact way to express repeated multiplication. Taking \( 10^{-3} \) as an example, it means \( 1 \) divided by \( 10 \) multiplied by itself three times. Therefore, \( 10^{-3} = \frac{1}{10 \times 10 \times 10} = \frac{1}{1000} \).

Important notes about exponents include:

• Positive exponents represent regular multiplication (e.g., \( 10^3 = 1000 \)).
• Negative exponents represent division of 1 by the base raised to the positive exponent (e.g., \( 10^{-3} = \frac{1}{1000} \)).

Understanding exponents is crucial in dealing with logarithms, as they are two sides of the same mathematical coin, where each depends on the manipulation of powers to define their relationship with numbers.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of logarithmic equations like \( \log_{10} x = -3 \), solving for \( x \) means determining what \( x \) must be for the expression to balance.

Here are the general steps to solve this type of equation:

• Understand the logarithmic equation and rewrite it using exponents; in this case, \( x = 10^{-3} \).
• Simplify the expression; calculate \( 10^{-3} \) to determine the specific numeric value of \( x \).
• Match the calculated solution with the multiple-choice answers given in the problem.

Through these steps, problem-solving becomes a structured approach of transforming and simplifying equations until the solution is evident. This systematic technique not only builds strong mathematical skills but also reinforces comprehension of core concepts like exponents and logarithms.