Problem 34 - Algebra 2 and Trigonometry

Question

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=-3 $$

Step-by-Step Solution

Answer

The antilogarithm is 0.0010.

Step 1: Understanding the Logarithm

The given equation is \( \log x = -3 \). This equation indicates that the logarithm of \(x\) with base 10 is -3.

Step 2: Converting to Exponential Form

To find the antilogarithm, we need to convert the logarithmic equation to its exponential form. The equation \( \log x = -3 \) can be rewritten in exponential form as \( x = 10^{-3} \).

Step 3: Calculating the Antilogarithm

Now, calculate \( 10^{-3} \) to find \(x\). This is done by computing \( x = \frac{1}{10^3} = \frac{1}{1000} \).

Step 4: Finding the Decimal Value

Compute \( \frac{1}{1000} \) to find its decimal form. Thus, \( x = 0.001 \). Since the problem asks for four decimal places, we confirm that \(0.0010\) is the correct answer in this format.

Key Concepts

Understanding Logarithms

Logarithms are the inverse operation of exponentiation. They answer the question: "To which number must we raise a base number to produce another number?" In simple terms, if you have a logarithmic expression like \( \log_b a = c \), it translates to "b raised to the power of c equals a."
Logarithms are used in many fields, such as science, engineering, and finance, for simplifying complex multiplication into addition, among other functions. In our exercise, we encounter a logarithmic statement: \( \log x = -3 \), where the base is implied to be 10.

• The base of the logarithm is the number you repeatedly multiply.
• The result of the logarithm gives the power to which the base is raised.

Here, "\( \log x = -3 \)" suggests that "What power must 10 be raised to in order to get x?" This will lead us to our next concept, exponential form.

From Logarithms to Exponential Form

Exponential form is a way to express operations involving numbers raised to a power. By converting logarithmic expressions to exponential form, we can more easily compute or understand them.
In our example \( \log x = -3 \), we need to convert this into its exponential counterpart. This involves rearranging the expression so that it reads \( x = 10^{-3} \). This change of form is crucial as it allows us to directly calculate the antilogarithm, or the original number prior to applying the logarithm.

• The base number becomes the base of the exponent in the exponential form.
• The result of the logarithm (\(-3\) in this case) becomes the exponent.

By doing this transformation, we are ready to compute the value of \( x \) more straightforwardly. Let's explore this computation next.

Calculating with Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, use 10 as the base, which makes them especially manageable.
Calculating the antilogarithm in our exercise involves dealing with a base 10 in the equation \( x = 10^{-3} \). This implies evaluating the expression \( 10^{-3} \), which is the same as \( \frac{1}{10^3} \) or \( \frac{1}{1000} \).

• Base 10 is intuitive because it mirrors our decimal number system, making it easy to understand and compute.
• Antilogarithms reverse the logarithmic operation, leading us to the original number.

Our calculation shows that \( x = 0.001 \) or written in four decimal places as \( 0.0010 \). This technique highlights the power of converting logarithmic equations into exponential form for simpler calculation.