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Chapter 9: Problem 63

Given \(\log _{b} 3=0.792\) and \(\log _{b} 5=1.161 .\) If possible, use the properties of logarithms to calculate values for each of the following. $$\log _{b} \frac{1}{5}$$

Short Answer

Expert verified
\( \log_{b} \frac{1}{5} = -1.161 \)

Step by step solution

01

Recall the property of logarithms

The logarithm of the reciprocal of a number can be found using the property \( \log_{b} \frac{1}{x} = -\log_{b} x \).
02

Apply the property to the given logarithm

Given \( \log _{b} 5=1.161 \), we can use the property from Step 1 to find \( \log_{b} \frac{1}{5} \): \( \log_{b} \frac{1}{5} = -\log_{b} 5 \).
03

Substitute the given value

Substitute the given value of \( \log _{b} 5 \): \( \log_{b} \frac{1}{5} = -1.161 \).

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Key Concepts

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

headline of the respective core concept
Logarithms are mathematical functions that tell us the power to which a base number must be raised to get another number. The notation is typically written as \(\text{log}_b (x)\), where \(b\) is the base and \(x\) is the number you're taking the logarithm of. For example, \(\text{log}_2 (8) = 3\) because \(2^3 = 8\).
Logarithms have several properties that make calculations easier, such as the product, quotient, and power rules. These properties can simplify complicated problems into manageable steps.
In this exercise, we focus on the reciprocal property to calculate \(\text{log}_b \frac{1}{5} \).
Reciprocal Property
The reciprocal property is one of the key properties of logarithms. It states that the logarithm of the reciprocal of a number is the negative of the logarithm of the number itself. Mathematically, it is written as \( \text{log}_b \frac{1}{x} = - \text{log}_b x \).
This means if you know \( \text{log}_b x \), you can easily find \( \text{log}_b \frac{1}{x} \) by simply negating the value.
This property is useful when dealing with fractions and can simplify otherwise complex logarithmic calculations.
Logarithmic Calculations
Logarithmic calculations often use various properties to simplify the process. Here鈥檚 how we can solve the given exercise step by step:
  • Step 1: Identify the given values and the property to use. Given that \( \text{log}_b 5 = 1.161 \), and we need to compute \( \text{log}_b \frac{1}{5} \).
  • Step 2: Use the reciprocal property. This property tells us that \( \text{log}_b \frac{1}{x} = - \text{log}_b x \).
  • Step 3: Substitute the known value into the property. We have \( \text{log}_b 5 = 1.161 \), so \( \text{log}_b \frac{1}{5} = -1.161 \).
By following these simple steps, we leverage logarithmic properties to make calculations straightforward and accurate.

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Most popular questions from this chapter

Find each of the following, given that $$ f(x)=\frac{1}{x+2} \quad \text { and } \quad g(x)=5 x-8 $$ $$ f(-1) $$

Simplify. $$ \log _{1 / 4} \frac{1}{64} $$

Let \(f(x)=\sqrt[3]{x}-4 .\) Use composition of inverse functions to show that $$f^{-1}(x)=x^{3}+4$$

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