Problem 95 - 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. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$. \log 1000$$

Step-by-Step Solution

Answer

\(\log_{10} 1000 = 3\)

Step 1: Analyze the Table

Start by closely observing the values provided in the table. Specifically, we are looking for where \(f(x) = 1000\).

Step 2: Identify the answer

Looking through the values of \(f(x)\), it can be seen that \(f(x) = 10^x = 1000\) when \(x = 3\).

Step 3: Interpret the result

This implies that \(\log_{10} 1000 = 3\), since logarithm is the inverse operation of exponentiation. This shows us how to read logarithmic values indirectly from the exponential function table.

Key Concepts

Exponents and Logarithms

Exponents and logarithms are fundamental concepts in mathematics that are deeply interrelated. In essence, an exponent refers to the number of times a number, called the base, is multiplied by itself. For example, in the expression 2 to the power of 3 (written as \(2^3\)), the base is 2, and it is multiplied by itself 3 times: \(2 \times 2 \times 2 = 8\).

Logarithms, on the other hand, are the inverse operation of exponentiation. They answer the question: 'To what power must we raise a base to obtain a certain number?' If we take the logarithm base 10 of 100, denoted as \(\log_{10} 100\), we are asking 'to what power must 10 be raised to get 100?' The answer is 2, because \(10^2 = 100\). Understanding this relationship is key when learning about these topics.

Function's Table of Values

A function's table of values is a powerful tool for understanding and visualizing the relationship between inputs and outputs of a function. The table typically lists pairs of numbers, with the first number representing the input (or \(x\)-value) and the second number representing the corresponding output (or the function's value \(f(x)\)) upon applying the function to the input.

For the given function \(f(x) = 10^x\), a table of values would list various exponents \(x\) alongside their resulting powers of 10. Such a table can not only help us find specific outputs for given inputs but also aids us in spotting patterns or behaviors of the function across a range of values.

Inverse Operations

In mathematics, inverse operations are pairs of operations that reverse the effects of each other. Addition and subtraction are classic examples, as are multiplication and division. When it comes to exponents and logarithms, these too are inverse operations.

If you have an equation like \(10^x = y\), then taking the logarithm base 10 of y will give you x: \(\log_{10} y = x\). This is because the logarithm function essentially 'undoes' the effect of the exponentiation. When applied to a function's table of values, understanding inverse operations allows us to move back and forth between exponents and logarithms, translating from the language of growth and decay to the language of powers and indices.

Interpreting Logarithms

Interpreting logarithms can initially be challenging, but with practice, it becomes a more intuitive process. To interpret a logarithm, think of it as asking a question: the expression \(\log_{b} A = x\) is asking 'To what power must the base \(b\) be raised to produce \(A\)?'. The answer is x.

To find \(\log 1000\) using the table of values for the function \(f(x)=10^x\), we identify the entry where the output is 1000. The input corresponding to this output gives us the logarithm. Thus, interpreting logarithms involves correlating the power and the result of exponentiation, and this understanding is crucial when working with logarithmic functions in any context.