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Chapter 4: Problem 17

$$ \begin{array}{l} \text { Given } \log _{b} 3=1.099 \text { and } \log _{b} 5=1.609, \text { find each }\\\ \text { of the following. } \end{array} $$ $$ \log _{b} 15 $$

Short Answer

Expert verified
\( \log_b 15 = 2.708 \).

Step by step solution

01

Understand the Problem

We are given \( \log_b 3 = 1.099 \) and \( \log_b 5 = 1.609 \). We need to find \( \log_b 15 \).
02

Use Product Rule of Logarithms

Recall the product rule for logarithms: \( \log_b(mn) = \log_b(m) + \log_b(n) \). In this case, we can write 15 as a product of 3 and 5.
03

Apply the Product Rule

Using the product rule, we get \( \log_b 15 = \log_b(3 \cdot 5) = \log_b 3 + \log_b 5 \).
04

Substitute Given Values

Substitute \( \log_b 3 \) and \( \log_b 5 \) with the given values: \( \log_b 15 = 1.099 + 1.609 \).
05

Perform the Addition

Add the values together: \( 1.099 + 1.609 = 2.708 \).
06

Write Final Answer

Therefore, \( \log_b 15 = 2.708 \).

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Key Concepts

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

logarithm
A logarithm is a mathematical function that answers the question: 'To what exponent must a base number be raised, to get another number?' In simpler terms, a logarithm helps you find the power to which a base number is raised to obtain a specific value. For example, if we say \(\text{log}_b(x) = y\), we mean that the base \(b\) raised to the power \(y\) equals \(x\).

Here's a quick example:
  • Consider the logarithm \(\text{log}_2(8) = 3\). This means that \(2\) raised to the power of \(3\) equals \(8\).
Logarithms are incredibly useful in various fields, such as science, engineering, and finance, because they can simplify complex calculations, especially those involving growth rates and exponential relationships.
logarithmic properties
Logarithms have several important properties that make calculations easier. One key property is the product rule:
\[ \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n) \] This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors.

Now, let's examine how to use this in our exercise:
  • Given \(\text{log}_b(3) = 1.099\) and \(\text{log}_b(5) = 1.609\), we need to find \(\text{log}_b(15)\).
  • We can write 15 as a product of 3 and 5. So, \( 15 = 3 \times 5 \).
Applying the product rule:
\[ \text{log}_b(15) = \text{log}_b(3 \times 5) = \text{log}_b(3) + \text{log}_b(5) \] By substituting the given values: \( \text{log}_b(15) = 1.099 + 1.609 = 2.708 \).

This property is invaluable for breaking down more complicated logarithmic expressions into simpler parts.
solving logarithmic equations
Solving logarithmic equations involves using logarithmic properties to simplify and solve for the unknown variable. Here鈥檚 a step-by-step approach:
  • Step 1: Isolate the logarithmic term if necessary.
  • Step 2: Use logarithmic properties, such as the product, quotient, or power rules, to combine or split the logarithms.
  • Step 3: If possible, rewrite the logarithmic equation in exponential form to solve for the variable.
Let's use these steps in an example:
  • Given: \(\text{log}_b(3) = 1.099\) and \(\text{log}_b(5) = 1.609\).
  • To find: \(\text{log}_b(15)\).
By applying the product rule: \( \text{log}_b(15) = \text{log}_b(3 \times 5) = \text{log}_b(3) + \text{log}_b(5) \)

Next, substitute the given values: \( \text{log}_b(15) = 1.099 + 1.609 \).

Finally, perform the addition: \( 1.099 + 1.609 = 2.708 \).

The logarithm of 15 with base \( b \) is 2.708.

Keeping these steps in mind will help you solve any logarithmic equation more efficiently and with greater confidence.

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