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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]
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