Problem 35 - Algebra 2 and Trigonometry

Question

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=-4 $$

Step-by-Step Solution

Answer

The antilogarithm is 0.0001.

Step 1: Understanding Logarithms and Antilogarithms

The problem provides a logarithmic equation, \( \log x = -4 \). In this context, \(x\) is the antilogarithm, which means we need to find the value of \(x\) that satisfies this equation. The logarithm equation is based on the common logarithm (base 10).

Step 2: Rewriting the Logarithmic Expression

From the equation \(\log x = -4\), we need to express \(x\) in terms of the base-10 antilogarithm. This means \(x\) can be expressed as an exponential function: \(x = 10^{-4}\).

Step 3: Calculating the Antilogarithm

Now calculate the value of \(x\) using the expression derived: \(x = 10^{-4}\). This calculation gives us \(x = 0.0001\).

Step 4: Rounding to Four Decimal Places

The result of the calculation, \(x = 0.0001\), is already to four decimal places. Therefore, no additional rounding is needed. The final value of \(x\) is 0.0001.

Key Concepts

Logarithms

Logarithms might sound complex, but at their core, they're just a way to represent exponents. If you've ever worked with exponential expressions, you're already on the right track. Logarithms essentially ask the question: "To what power must we raise a certain base to get a specific number?" Commonly, you'll encounter logarithms with base 10, often denoted as "log" without a base number written. When you see something like \( \log x = -4 \), it means that 10 raised to the power of -4 equals \( x \).
Understanding this relationship is crucial because it transforms multiplicative processes into more manageable additive ones, simplifying many calculations. Logarithms are invaluable in fields such as engineering, science, and statistics. They allow us to handle very large or very small numbers more easily and are foundational in operations involving exponentials and growth rates. So, next time you come across a logarithm, remember: it's just the exponent that base 10 needs to become the number you're interested in!

Exponential Functions

Exponential functions, like the one we used to find the antilogarithm, are mathematical expressions that get really big or really small depending on the sign of the exponential. The formula \( x = 10^{-4} \) comes in handy here, as it represents an exponential function of base 10. An exponential function generally looks like \( a^x \), where "a" is the base and "x" is the exponent.
These functions grow (or shrink) multiplicatively rather than linearly, which means doubling the exponent leads to a squaring of the outcome. This property is essential in modeling phenomena that involve fast growth or decay, such as population studies or radioactive decay.

• Positive exponents lead to repeated multiplication.
• Negative exponents translate to repeated division.

In practical terms, understanding exponential functions helps you handle problems involving change and transformation more intuitively. They are everywhere around us—from calculating interest on savings to determining half-lives of substances.

Base 10

The concept of base 10 is familiar—it’s the number system most of us use daily for everything from counting cash to timing our walks. But in mathematics, base 10 holds a deeper utility, especially concerning logs and antilogs. Logarithms can have any number as a base, but base 10 is the most common due to its direct correlation to our decimal counting system.
Working with base-10 simplifies converting log solutions into numbers we can easily interpret and use, such as converting \( \log x = -4 \) into the exponential form \( x = 10^{-4} \). Base 10 logarithms are also easier to interpret because divisor factors of 10 are simpler to break down mentally. This is especially true in scientific contexts, where values often span many orders of magnitude.

• Helps interpret large and small numbers effortlessly.
• Encourages intuitive understanding through daily relevance.

So grasping the significance of base 10 is not only necessary for tackling logarithmic functions but also advantageous for simplifying complex numerical problems.