At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Let's solve the given problem thoroughly and step by step to determine which of the given expressions are equivalent to the expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
### Step 1: Simplify the given expression
We start with the given expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
Using the properties of logarithms:
[tex]\[ \log_b (mn) = \log_b m + \log_b n \][/tex]
[tex]\[ \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \][/tex]
[tex]\[ \log_b (m^n) = n \log_b m \][/tex]
First, note that [tex]\(\log_{10} 10 = 1\)[/tex], so the expression simplifies as follows:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - 1 \][/tex]
Next, rewrite [tex]\(\log_{10} 20\)[/tex]:
[tex]\[ \log_{10} 20 = \log_{10} (2 \cdot 10) = \log_{10} 2 + \log_{10} 10 = \log_{10} 2 + 1 \][/tex]
Replace this in the simplified expression:
[tex]\[ 5 \log_{10} x + (\log_{10} 2 + 1) - 1 \][/tex]
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
The final simplified form is:
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
### Step 2: Rewrite using logarithmic properties
Using the logarithmic property [tex]\(\log_b (mn) = \log_b m + \log_b n\)[/tex], we combine the two logarithms:
[tex]\[ 5 \log_{10} x + \log_{10} 2 = \log_{10} (x^5) + \log_{10} 2 = \log_{10} (2 x^5) \][/tex]
So, our given expression is equivalent to:
[tex]\[ \log_{10} (2 x^5) \][/tex]
### Step 3: Check all given expressions
- Expression 1: [tex]\(\log_{10} (2 x^5)\)[/tex]
This matches exactly with our simplified form above. So, this is indeed equivalent.
- Expression 2: [tex]\(\log_{10} (20 x^5) - 1\)[/tex]
Rewrite [tex]\(\log_{10} (20 x^5)\)[/tex]:
[tex]\[ \log_{10} (20 x^5) = \log_{10} (20) + \log_{10} (x^5) = \log_{10} 20 + 5 \log_{10} x \][/tex]
Using [tex]\(\log_{10} 20 = \log_{10} (2 \cdot 10) = \log_{10} 2 + \log_{10} 10 = \log_{10} 2 + 1\)[/tex], we get:
[tex]\[ \log_{10} (20 x^5) = (\log_{10} 2 + 1) + 5 \log_{10} x = \log_{10} 2 + 5 \log_{10} x + 1 \][/tex]
Subtracting 1:
[tex]\[ \log_{10} 2 + 5 \log_{10} x + 1 - 1 = \log_{10} 2 + 5 \log_{10} x \][/tex]
Which is equal to:
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
This matches our simplified form, so it is also equivalent.
- Expression 3: [tex]\(\log_{10} (100 x) + 1\)[/tex]
Rewrite [tex]\(\log_{10} (100 x)\)[/tex]:
[tex]\[ \log_{10} (100 x) = \log_{10} 100 + \log_{10} x = 2 + \log_{10} x \][/tex]
Adding 1:
[tex]\[ 2 + \log_{10} x + 1 = 3 + \log_{10} x \][/tex]
This does not match our simplified form, so it is not equivalent.
- Expression 4: [tex]\(\log_{10} (2 x)^5\)[/tex]
Using the logarithmic property:
[tex]\[ \log_{10} (2 x)^5 = 5 \log_{10} (2 x) = 5 (\log_{10} 2 + \log_{10} x) = 5 \log_{10} 2 + 5 \log_{10} x \][/tex]
This does not match our simplified form, so it is not equivalent.
- Expression 5: [tex]\(\log_{10} (10 x)\)[/tex]
Rewrite [tex]\(\log_{10} (10 x)\)[/tex]:
[tex]\[ \log_{10} (10 x) = \log_{10} 10 + \log_{10} x = 1 + \log_{10} x \][/tex]
This does not match our simplified form, so it is not equivalent.
### Final Answer
The equivalent expressions to the given expression are:
1. [tex]\(\log_{10} (2 x^5)\)[/tex]
2. [tex]\(\log_{10} (20 x^5) - 1\)[/tex]
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
### Step 1: Simplify the given expression
We start with the given expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
Using the properties of logarithms:
[tex]\[ \log_b (mn) = \log_b m + \log_b n \][/tex]
[tex]\[ \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \][/tex]
[tex]\[ \log_b (m^n) = n \log_b m \][/tex]
First, note that [tex]\(\log_{10} 10 = 1\)[/tex], so the expression simplifies as follows:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - 1 \][/tex]
Next, rewrite [tex]\(\log_{10} 20\)[/tex]:
[tex]\[ \log_{10} 20 = \log_{10} (2 \cdot 10) = \log_{10} 2 + \log_{10} 10 = \log_{10} 2 + 1 \][/tex]
Replace this in the simplified expression:
[tex]\[ 5 \log_{10} x + (\log_{10} 2 + 1) - 1 \][/tex]
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
The final simplified form is:
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
### Step 2: Rewrite using logarithmic properties
Using the logarithmic property [tex]\(\log_b (mn) = \log_b m + \log_b n\)[/tex], we combine the two logarithms:
[tex]\[ 5 \log_{10} x + \log_{10} 2 = \log_{10} (x^5) + \log_{10} 2 = \log_{10} (2 x^5) \][/tex]
So, our given expression is equivalent to:
[tex]\[ \log_{10} (2 x^5) \][/tex]
### Step 3: Check all given expressions
- Expression 1: [tex]\(\log_{10} (2 x^5)\)[/tex]
This matches exactly with our simplified form above. So, this is indeed equivalent.
- Expression 2: [tex]\(\log_{10} (20 x^5) - 1\)[/tex]
Rewrite [tex]\(\log_{10} (20 x^5)\)[/tex]:
[tex]\[ \log_{10} (20 x^5) = \log_{10} (20) + \log_{10} (x^5) = \log_{10} 20 + 5 \log_{10} x \][/tex]
Using [tex]\(\log_{10} 20 = \log_{10} (2 \cdot 10) = \log_{10} 2 + \log_{10} 10 = \log_{10} 2 + 1\)[/tex], we get:
[tex]\[ \log_{10} (20 x^5) = (\log_{10} 2 + 1) + 5 \log_{10} x = \log_{10} 2 + 5 \log_{10} x + 1 \][/tex]
Subtracting 1:
[tex]\[ \log_{10} 2 + 5 \log_{10} x + 1 - 1 = \log_{10} 2 + 5 \log_{10} x \][/tex]
Which is equal to:
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
This matches our simplified form, so it is also equivalent.
- Expression 3: [tex]\(\log_{10} (100 x) + 1\)[/tex]
Rewrite [tex]\(\log_{10} (100 x)\)[/tex]:
[tex]\[ \log_{10} (100 x) = \log_{10} 100 + \log_{10} x = 2 + \log_{10} x \][/tex]
Adding 1:
[tex]\[ 2 + \log_{10} x + 1 = 3 + \log_{10} x \][/tex]
This does not match our simplified form, so it is not equivalent.
- Expression 4: [tex]\(\log_{10} (2 x)^5\)[/tex]
Using the logarithmic property:
[tex]\[ \log_{10} (2 x)^5 = 5 \log_{10} (2 x) = 5 (\log_{10} 2 + \log_{10} x) = 5 \log_{10} 2 + 5 \log_{10} x \][/tex]
This does not match our simplified form, so it is not equivalent.
- Expression 5: [tex]\(\log_{10} (10 x)\)[/tex]
Rewrite [tex]\(\log_{10} (10 x)\)[/tex]:
[tex]\[ \log_{10} (10 x) = \log_{10} 10 + \log_{10} x = 1 + \log_{10} x \][/tex]
This does not match our simplified form, so it is not equivalent.
### Final Answer
The equivalent expressions to the given expression are:
1. [tex]\(\log_{10} (2 x^5)\)[/tex]
2. [tex]\(\log_{10} (20 x^5) - 1\)[/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
