Problem 98 - Precalculus : Building Concepts and Connections

Question

Explain how you could use the following table of values for the function \(f(x)=10^{x}\) to find the given quantity. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$\log \sqrt{10}$$

Step-by-Step Solution

Answer

The solution for the given quantity \(log \sqrt{10}\) is 0.5

Step 1: Understanding logarithmic function

This step is about understanding the nature of the logarithmic function being inversed to the exponentiation. One of the important properties of logarithm is that for any base 'b' , \(\log_{b} b^{x} = x\). This means that, for the function \(f(x)=10^{x}\), the logarithm of the function will bring back 'x'. Understanding this property will help to solve the exercise.

Step 2: Find the exponent for base 10

Looking at the given quantity \(log \sqrt{10}\), the task is to find the exponent 'x' for 10. The \(\sqrt{10}\) is the same as \(10^{0.5}\), because any square root can be expressed as an exponent of a half. So we replace \(log \sqrt{10}\) with \(log 10^{0.5}\)

Step 3: Apply the property of logarithm

Now apply the logarithm property \(\log_{b} b^{x} = x\), so we obtain \(log 10^{0.5} = 0.5\)

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in mathematics, where a constant base is raised to a variable exponent. In this problem, we're looking at the exponential function with base 10: \(f(x) = 10^x\). This means the function takes a number \(x\) and computes \(10\) raised to that power.

Exponential functions are widely used in various domains, including science, finance, and engineering, because they model growth processes, like population growth or compound interest. Here are some key aspects to remember about exponential functions:

• The base, in this case 10, remains constant.
• The exponent \(x\) determines the output of the function.
• A positive exponent typically results in larger numbers, while a negative exponent results in fractions.

In the provided table, you can observe different outputs for different values of \(x\). For instance, when \(x = 3\), the output is \(1000\), which relates to the idea that \(10^3 = 1000\). Understanding this growth behavior is crucial when working with exponential functions.

Logarithmic Properties

Logarithms are the inverse of exponential functions, which means they have the opposite operation. When working with base 10 logarithms, like \(\log_{10}(x)\), the objective is to find what power you need to raise 10 to get x.

Logarithmic properties simplify complex multiplication and division into manageable addition and subtraction. Here are some important properties of logarithms:

• Inverse Property: For a logarithm with a base, \(\log_{b}(b^x) = x\). This is the property that allows us to transform between the exponential function and its associated logarithm.
• Product Property: \(\log_{b}(XY) = \log_{b}(X) + \log_{b}(Y)\).
• Quotient Property: \(\log_{b}\left(\frac{X}{Y}\right) = \log_{b}(X) - \log_{b}(Y)\).
• Power Property: \(\log_{b}(X^y) = y \cdot \log_{b}(X)\).

These properties make it easier to manipulate expressions without computing large or cumbersome numbers directly. In the exercise, the expression \(\log \sqrt{10}\) was simplified using the power property, showing a clear application of these logarithmic rules.

Base 10

Base 10 logarithms, also known as common logarithms, are especially important because our number system is decimal-based. Many computations, particularly in sciences and engineering, use base 10 due to the ease of interpreting results.
Using base 10 is intuitive because it aligns with our everyday counting system. Here are several reasons why base 10 is preferred:

• Ease of Use: Base 10 relates closely to our decimal system, making it simpler for calculation and understanding.
• Widely Accepted: Many technologies, especially older calculators, use base 10 as their standard logarithm base, known as "log."
• Simplification: When working with numbers like 10, 100, 1000, etc., base 10 allows quick simplification as the logs become whole numbers.

In the solution given, the logarithmic problem was simplified using base 10, which made it straightforward to compute \(\log \sqrt{10}\) as \(0.5\). This shows how powerful and convenient base 10 logarithms can be for simplifying mathematical expressions.