Problem 97 - 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. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$\log 0.5$$

Step-by-Step Solution

Answer

\(\log 0.5 = -0.3010\)

Step 1: Understand the relationship between logarithmic and exponential functions

The relationship between the logarithmic and the exponential function is that they are inverse operations of each other. In other words, for the base \(10\), this means that \( \log_{10} (10^x) = x \) and \(10^{\log_{10}x} = x\). This relationship will be used in interpreting the table of values.

Step 2: Interpret the table based on the relationship

From the table, look for the case where \(f(x) = 0.5 \). It can be found that when \(f(x) = 0.5 \), \( x = -0.3010\). This implies that \(\log_{10} 0.5 = -0.3010\).

Step 3: State the Final Answer

Using the table of the function \(f(x) = 10^x\) and understanding the relationship between logarithms and exponential functions, the value of \(\log 0.5\) is found to be \(-0.3010\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. This base is a constant number, and the exponent is a variable that represents a power. This type of function is represented as \( f(x) = b^x \), where \( b \) is the base and \( x \) is the exponent. In the example provided in the exercise, the base is \( 10 \), so the function becomes \( f(x) = 10^x \).
Exponential functions are essential in various fields, such as biology, finance, and physics, because they describe processes that grow exponentially or change at constant percentage rates.
Some properties of exponential functions include:

• The function has no upper bound; it continues to increase as \( x \) increases.
• The function is always positive because any positive number to any power is greater than zero.
• The graph of an exponential function is a curve that rises continuously from left to right.

Understanding exponential functions is crucial for interpreting tables of values, especially when dealing with their inverse operations, like logarithms.

Inverse Operations

Inverse operations are pairs of operations that reverse each other. In arithmetic, addition and subtraction are inverse operations, and multiplication and division are also inverse operations. For exponential and logarithmic functions, they are inverses as well.
The essence of inverse operations is in their ability to "undo" one another. When you apply an operation and then its inverse, you return to your original value. For the exponential function \(10^x\) and the logarithm function \(\log_{10}(x)\), this means that \( \log_{10}(10^x) = x \) and \( 10^{\log_{10}(x)} = x \).

• This inverse property helps solve equations that include exponential terms by applying logarithms, which bring down exponents.
• Similarly, when dealing with logarithmic equations, the corresponding exponential function can help simplify and find solutions.

The table in the exercise leverages the inverse relationship, where knowledge of \( f(x) \) can be used to determine values of the inverse function \( \log_{10}(x)\). This makes it easier to handle complex exponential equations and find meaningful solutions.

Base 10 Logarithms

Base 10 logarithms, often referred to as common logarithms, are especially significant because they use the constant base of 10. They are widely used in scientific notation and practical applications because they simplify calculations with tens, hundreds, and thousands, aligning well with our decimal-based number system.
Understanding base 10 logarithms involves interpreting expressions of the form \( \log_{10}(x) \), which represent the power to which the base 10 must be raised to obtain \( x \). For example, if \( \log_{10}(100) = 2 \), it means that 10 must be raised to the power of 2 to equal 100.

• These logarithms help condense large and small numbers into manageable figures.
• They are key in solving exponential equations and converting multiplicative relationships into additive ones, which simplifies complex calculations.

In relation to the exercise, using base 10 logarithms allows for identifying specific powers of 10 in the table efficiently, as seen where \(\log_{10}(0.5)\) was derived as \(-0.3010\), indicating that raising 10 to the power of \(-0.3010\) equals 0.5. This clarity and ease of use make base 10 logarithms a staple in both theoretical and applied mathematics.