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Chapter 11: Problem 123

Explain the mathematical relationship between \(f(x)=\log x\) and \(g(x)=10^{x}\)

Short Answer

Expert verified
The functions are inverses: \(f(x) = \log x\) and \(g(x) = 10^x\) cancel each other out.

Step by step solution

01

Understanding the Functions

We have two functions: the logarithmic function \(f(x) = \log x\) and the exponential function \(g(x) = 10^x\). The base of the logarithm for \(f(x)\) is 10 as it is a simplified form of \(\log_{10} x\). Both functions involve exponentiation and logarithms, which are inverse operations of each other.
02

Identify the Inverse Relationship

The logarithmic function \(f(x) = \log x\) returns the exponent to which the base 10 must be raised to produce \(x\). Conversely, the exponential function \(g(x) = 10^x\) returns a number (\(10\) raised to the power \(x\)). This demonstrates that \(f(x)\) and \(g(x)\) are inverse functions.
03

Verify the Inverse Property

For \(f(x)\) and \(g(x)\) to be inverses, \(f(g(x))\) should equal \(x\), and \(g(f(x))\) should also equal \(x\).\1. \(f(g(x)) = \log(10^x) = x\) because exponentiation and logarithms are inverse operations.\2. \(g(f(x)) = 10^{\log x} = x\). Thus, the actions of \(f\) and \(g\) cancel each other, confirming their inverse relationship.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Function
In the world of mathematics, a logarithmic function is a powerful tool that helps us understand exponential relationships in reverse. The basic form of the logarithmic function we're discussing here is expressed as:
- \( f(x) = \log x \)
This notation is actually a shorthand for the base 10 logarithm, meaning it's often written as \( \log_{10} x \). When you see this function, it's asking the question: "What power must 10 be raised to in order to obtain \( x \)?"
Here are some essential things to remember about logarithmic functions:
  • They are used to solve for the exponent in equations where the result of an exponentiation is known, but the exponent is not.
  • In computational contexts, logarithms transform multiplication into addition, making complex calculations easier.
The logarithm helps us navigate the world of exponential scales, such as calculating decibels in sound intensity or the pH level in chemistry.
Exponential Function
The exponential function serves as the counterpart to our earlier logarithmic function. For this particular discussion, the function is defined as:
- \( g(x) = 10^x \)
This tells us that the function takes the exponential form where 10 is raised to the power of \( x \). Such functions model growth or decay processes, where quantities increase or decrease by consistent multiplicative factors.
  • Exponential functions grow remarkably fast. For example, even small changes in \( x \) can result in large changes in \( g(x) \).
  • They are used extensively in fields like biology (population growth), finance (compound interest), and computing (data growth).
The exponential function serves as a foundation for understanding rapid change and natural laws dictating compound growth or decay.
Base 10 Logarithm
The base 10 logarithm, also referred to as the common logarithm, is pivotal in various scientific and engineering domains. When the base of a logarithm is 10, this simplifies notation to \( \log x \) without requiring an explicit base indication. Here's why this base is particularly useful:
  • Human perception often scales logarithmically, making base 10 logarithms suitable for measuring quantities like sound intensity and earthquake magnitude.
  • They create a straightforward link between powers of 10, which are easy for people to understand given the decimal system used globally.
In essence, the base 10 logarithm acts as an intuitive tool for analyzing, simplifying, and interpreting phenomena that vary exponentially over a range of scales.

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Most popular questions from this chapter

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 29=5^{x-6} $$

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest tenth. See Using Your Calculator: Solving Exponential Equations Graphically or Solving Logarithmic Equations Graphically. $$ 3^{x}-10=3^{-x} $$

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ e^{4 x}=60 $$

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log \frac{2-5 x}{2(x+8)}=0 $$

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log (x+90)+\log x=3 $$
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