Answer :
To determine which expressions are equivalent to [tex]\(\log_3 81 + \log_3 81\)[/tex], let's proceed with the solution step-by-step.
### Step 1: Apply the Logarithm Addition Rule
The sum of two logarithms with the same base can be converted into a single logarithm of a product:
[tex]\[ \log_3 81 + \log_3 81 = \log_3 (81 \cdot 81) \][/tex]
### Step 2: Calculate the Product
Next, calculate the product inside the log:
[tex]\[ 81 \cdot 81 = 6561 \][/tex]
So we have:
[tex]\[ \log_3 (81 \cdot 81) = \log_3 6561 \][/tex]
### Step 3: Evaluate the Logarithm
We need to evaluate [tex]\(\log_3 6561\)[/tex]. Notice that:
[tex]\[ 6561 = 3^8 \][/tex]
Thus:
[tex]\[ \log_3 6561 = \log_3 (3^8) = 8 \][/tex]
Now let's compare the given expressions to our derived expression [tex]\(\log_3 81 + \log_3 81\)[/tex]:
### Reviewing the Options
A. [tex]\(8\)[/tex]:
- Yes, as we have shown, [tex]\(\log_3 81 + \log_3 81 = 8\)[/tex].
B. [tex]\(\log_3 (3^8)\)[/tex]:
- Yes, since [tex]\(3^8 = 6561\)[/tex], and we have [tex]\(\log_3 (3^8) = 8\)[/tex].
C. [tex]\(\log 6561\)[/tex]:
- No, [tex]\(\log 6561\)[/tex] typically means [tex]\(\log_{10} 6561\)[/tex] (logarithm base 10), which is not equivalent to [tex]\(\log_3 6561\)[/tex]. Thus, this is not equivalent.
D. [tex]\(\log_3 6561\)[/tex]:
- Yes, [tex]\(\log_3 6561\)[/tex] is exactly what we derived above to be equivalent to [tex]\(\log_3 81 + \log_3 81\)[/tex].
### Conclusion
Hence, the expressions that are equivalent to [tex]\(\log_3 81 + \log_3 81\)[/tex] are:
- A. 8
- B. [tex]\(\log_3 (3^8)\)[/tex]
- D. [tex]\(\log_3 6561\)[/tex]
### Step 1: Apply the Logarithm Addition Rule
The sum of two logarithms with the same base can be converted into a single logarithm of a product:
[tex]\[ \log_3 81 + \log_3 81 = \log_3 (81 \cdot 81) \][/tex]
### Step 2: Calculate the Product
Next, calculate the product inside the log:
[tex]\[ 81 \cdot 81 = 6561 \][/tex]
So we have:
[tex]\[ \log_3 (81 \cdot 81) = \log_3 6561 \][/tex]
### Step 3: Evaluate the Logarithm
We need to evaluate [tex]\(\log_3 6561\)[/tex]. Notice that:
[tex]\[ 6561 = 3^8 \][/tex]
Thus:
[tex]\[ \log_3 6561 = \log_3 (3^8) = 8 \][/tex]
Now let's compare the given expressions to our derived expression [tex]\(\log_3 81 + \log_3 81\)[/tex]:
### Reviewing the Options
A. [tex]\(8\)[/tex]:
- Yes, as we have shown, [tex]\(\log_3 81 + \log_3 81 = 8\)[/tex].
B. [tex]\(\log_3 (3^8)\)[/tex]:
- Yes, since [tex]\(3^8 = 6561\)[/tex], and we have [tex]\(\log_3 (3^8) = 8\)[/tex].
C. [tex]\(\log 6561\)[/tex]:
- No, [tex]\(\log 6561\)[/tex] typically means [tex]\(\log_{10} 6561\)[/tex] (logarithm base 10), which is not equivalent to [tex]\(\log_3 6561\)[/tex]. Thus, this is not equivalent.
D. [tex]\(\log_3 6561\)[/tex]:
- Yes, [tex]\(\log_3 6561\)[/tex] is exactly what we derived above to be equivalent to [tex]\(\log_3 81 + \log_3 81\)[/tex].
### Conclusion
Hence, the expressions that are equivalent to [tex]\(\log_3 81 + \log_3 81\)[/tex] are:
- A. 8
- B. [tex]\(\log_3 (3^8)\)[/tex]
- D. [tex]\(\log_3 6561\)[/tex]
