Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To expand the logarithmic expression [tex]\(\log_7(6st)\)[/tex] using the Laws of Logarithms, follow these steps:
### Step-by-Step Solution:
1. Understand the Product Rule for Logarithms:
The product rule for logarithms states that:
[tex]\[ \log_b(m \cdot n) = \log_b(m) + \log_b(n) \][/tex]
This rule applies when you have a product inside the logarithm and allows you to separate it into the sum of individual logarithms.
2. Identify the parts inside the logarithm:
The expression inside the logarithm is [tex]\(6st\)[/tex]. This is a product of three terms: 6, [tex]\(s\)[/tex], and [tex]\(t\)[/tex].
3. Apply the Product Rule iteratively:
First, separate the logarithm of the product of 6 and [tex]\(st\)[/tex]:
[tex]\[ \log_7(6st) = \log_7(6) + \log_7(st) \][/tex]
Next, apply the product rule again to the [tex]\(\log_7(st)\)[/tex] term:
[tex]\[ \log_7(st) = \log_7(s) + \log_7(t) \][/tex]
4. Combine all the separated terms:
Putting it all together, we have:
[tex]\[ \log_7(6st) = \log_7(6) + \log_7(s) + \log_7(t) \][/tex]
### Final Expanded Expression:
[tex]\[ \log_7(6) + \log_7(s) + \log_7(t) \][/tex]
Thus, the expanded form of the logarithmic expression [tex]\(\log_7(6st)\)[/tex] is:
[tex]\[ \log_7(6) + \log_7(s) + \log_7(t) \][/tex]
### Step-by-Step Solution:
1. Understand the Product Rule for Logarithms:
The product rule for logarithms states that:
[tex]\[ \log_b(m \cdot n) = \log_b(m) + \log_b(n) \][/tex]
This rule applies when you have a product inside the logarithm and allows you to separate it into the sum of individual logarithms.
2. Identify the parts inside the logarithm:
The expression inside the logarithm is [tex]\(6st\)[/tex]. This is a product of three terms: 6, [tex]\(s\)[/tex], and [tex]\(t\)[/tex].
3. Apply the Product Rule iteratively:
First, separate the logarithm of the product of 6 and [tex]\(st\)[/tex]:
[tex]\[ \log_7(6st) = \log_7(6) + \log_7(st) \][/tex]
Next, apply the product rule again to the [tex]\(\log_7(st)\)[/tex] term:
[tex]\[ \log_7(st) = \log_7(s) + \log_7(t) \][/tex]
4. Combine all the separated terms:
Putting it all together, we have:
[tex]\[ \log_7(6st) = \log_7(6) + \log_7(s) + \log_7(t) \][/tex]
### Final Expanded Expression:
[tex]\[ \log_7(6) + \log_7(s) + \log_7(t) \][/tex]
Thus, the expanded form of the logarithmic expression [tex]\(\log_7(6st)\)[/tex] is:
[tex]\[ \log_7(6) + \log_7(s) + \log_7(t) \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.