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Chapter 3: Problem 21

Given \(\log _{b} 3=1.099\) and \(\log _{b} 5=1.609,\) find each value. $$\log _{b}(5 b)$$

Short Answer

Expert verified
The value of \log_{b}(5b) is 2.609.\

Step by step solution

01

- Apply the Product Rule

The logarithm of a product can be expressed as a sum of logarithms. Use the product rule: \[\text{If } \log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\]For our problem, let \(x=5\) and \(y=b\)
02

- Substitute the Given Values

Substitute \(x=5\) and \(y=b\) in the product rule:\[\text{\log_{b}(5b) = \log_{b}(5) + \log_{b}(b) }\]Given that \log_{b}(5)=1.609\.
03

- Simplify the Expression

Recall that \(\log_{b}(b) = 1\), because any number raised to the power of its own base is 1:\[\log_{b} b = 1\]Therefore, we can rewrite the expression as:\[\log_{b}(5b) = 1.609 + 1\]
04

- Calculate the Sum

Add the values inside the logarithm expression:\[1.609 + 1 = 2.609\]Thus, \log_{b}(5b) = 2.609\.

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Key Concepts

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

Product Rule in Logarithms
Understanding how to simplify logarithms using rules is crucial. The product rule for logarithms helps us break down the logarithm of a product into the sum of individual logarithms. This rule states that for any positive numbers \(x\), \(y\), and base \(b\):
\(\text{If } \log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\)
For example, in the given problem, we need to find \(\log_{b}(5b)\). By applying the product rule, we break it down to \(\log_{b}(5)\) and \(\log_{b}(b)\). This simplifies our calculation, making it easier to find the solution.
Logarithm Properties
Logarithms have several properties that simplify complex expressions. These properties include:
  • Product Rule: As previously mentioned, \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\)
  • Quotient Rule: This lets us express the logarithm of a quotient as the difference of logarithms: \(\log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y)\)
  • Power Rule: This rule states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number: \(\log_{b}(x^y) = y \log_{b}(x)\)
  • Change of Base Formula: This is used when we need to change the base of the logarithm: \(\log_{b}(x) = \frac{\log_{k}(x)}{\log_{k}(b)}\)
These properties save time and effort, allowing us to simplify complex logarithmic expressions like in this exercise.
Base of Logarithm
In logarithms, the base is the number that you repeatedly multiply to get another number. The notation \(\log_{b}(x)\) means the logarithm of \(x\) with base \(b\). A few key points about the base:
  • Common Logarithm: When the base is 10, it is called the common logarithm. It is often written simply as \(\log(x)\).
  • Natural Logarithm: When the base is the number \(e\) (approximately 2.718), it is called the natural logarithm, denoted as \(\ln(x)\).
  • Base \(b\): Any positive number can be used as a base, but it must remain consistent throughout the problem.

For example, in our exercise, we dealt with a logarithm of base \(b\). Recall that \(\log_{b}(b) = 1\), which is a fundamental property. This simplification helps in breaking down the expression \(\log_{b}(5b)\) to arrive at the final solution.

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Most popular questions from this chapter

The percentage \(P\) of doctors who prescribe a certain new medicine is \(P(t)=100\left(1-e^{-0.2 t}\right)\) where \(t\) is the time, in months. a) Find \(P(1)\) and \(P(6)\) b) Find \(P^{\prime}(t)\) c) How many months will it take for \(90 \%\) of doctors to prescribe the new medicine? d) Find \(\lim _{t \rightarrow \infty} P(t),\) and discuss its meaning.

Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given. $$f(x)=3^{\left(2^{x}\right)}$$

Differentiate. $$g(x)=\sqrt{x^{3}-x}\left(\log _{5} x\right)$$

Differentiate. $$y=\ln \sqrt{5+x^{2}}$$

Differentiate. $$f(x)=\frac{1}{5} x^{5}\left(\ln x-\frac{1}{5}\right)$$
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