/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 107 If \(f(x)=\log _{a} x,\) show th... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 107

If \(f(x)=\log _{a} x,\) show that \(f\left(\frac{1}{x}\right)=-f(x)\)

Short Answer

Expert verified
Since \(f\left(\frac{1}{x}\right) = -\log_{a}(x) = -f(x)\), we have \(f\left(\frac{1}{x}\right) = -f(x)\).

Step by step solution

01

- Understand the given function

Given the function is:ewline\[ f(x) = \log_{a}(x) \] This function represents the logarithm of \(x\) with base \(a\).
02

- Substitute \( \frac{1}{x} \) into the function

We need to find the value of \[ f \left( \frac{1}{x} \right) \] Substitute \( \frac{1}{x} \) into the function \[ f\left(\frac{1}{x}\right) = \log_{a} \left( \frac{1}{x} \right) \]
03

- Apply the properties of logarithms

Use the property of logarithms that \( \log_{a} \left( \frac{1}{x} \right) = -\log_{a} (x) \), which states the logarithm of the reciprocal of a number is the negative logarithm of the number:\[ f \left( \frac{1}{x} \right) = -\log_{a} (x) \]
04

- Conclude the result

Since \[ -\log_{a} (x) = -f (x) \], we can write:\[ f\left(\frac{1}{x}\right) = -f(x) \]

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Key Concepts

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

logarithms
Logarithms are mathematical functions that help us to simplify and solve exponential equations. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, if we have \(\text{log}_{b}(a) = c\), it means that \(b^{c} = a\). Here, \(b\) is the base, \(a\) is the number, and \(c\) is the logarithm. Understanding logarithms is crucial in many areas of math and science, especially when dealing with exponential growth or decay.
logarithm of a reciprocal
The logarithm of a reciprocal involves taking the logarithm of the fraction \(\frac{1}{x}\). According to the properties of logarithms, this can be simplified using a specific rule: the logarithm of a reciprocal is equal to the negative logarithm of the original number. Mathematically, this is represented as \(\text{log}_{a}(\frac{1}{x}) = -\text{log}_{a}(x)\). This property can be extremely helpful when solving logarithmic equations involving reciprocals.
algebraic proofs
Algebraic proofs involve logical steps and properties of algebra to demonstrate a mathematical statement is true. To prove the properties of functions or equations, you often manipulate the expressions using known algebraic rules and properties. For instance, in the given problem, we used the properties of logarithms to prove the relationship \(f\big(\frac{1}{x}\big) = -f(x)\). By substituting \(\frac{1}{x} \) into the logarithmic function and using the property of reciprocal logarithms, we reached the desired conclusion.
properties of logarithms
Properties of logarithms are essential tools that make calculations more straightforward. Some key properties include:
  • \(\text{log}_{a}(x \times y) = \text{log}_{a}(x) + \text{log}_{a}(y)\)
  • \(\text{log}_{a}(\frac{x}{y}) = \text{log}_{a}(x) - \text{log}_{a}(y)\)
  • \(\text{log}_{a}(x^{y}) = y \times \text{log}_{a}(x)\)
  • \(\text{log}_{a}(\frac{1}{x}) = -\text{log}_{a}(x)\)

These properties allow simplification and easy manipulation of logarithmic expressions, making it easier to solve complex equations.

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Most popular questions from this chapter

\(\log _{2}(x+3)=2 \log _{2}(x-3)\)

Solve each equation. $$ \log _{3}(3 x-2)=2 $$

Graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{2}(x+2)\)

Time to Double or Triple an Investment The formula $$ t=\frac{\ln m}{n \ln \left(1+\frac{r}{n}\right)} $$ can be used to find the number of years \(t\) required to multiply an investment \(m\) times when \(r\) is the per annum interest rate compounded \(n\) times a year. (a) How many years will it take to double the value of an IRA that compounds annually at the rate of \(6 \% ?\) (b) How many years will it take to triple the value of a savings account that compounds quarterly at an annual rate of \(5 \% ?\) (c) Give a derivation of this formula.

Analyzing Interest Rates on a Mortgage Colleen and Bill have just purchased a house for 650,000, with the seller holding a second mortgage of 100,000 . They promise to pay the seller 100,000 plus all accrued interest 5 years from now. The seller offers them three interest options on the second mortgage: (a) Simple interest at 6 % per annum (b) 5.5 % interest compounded monthly (c) $5.25 % interest compounded continuously Which option is best? That is, which results in paying the least interest on the loan?
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