Answer :
I'd be happy to help you with that!
**a. log₃(140)**
To evaluate this logarithm, we can use the change of base formula:
log₃(x) = log(e) / log(e)/3
where e is the base of the natural logarithm.
log₃(140) = log(e) / log(3)
First, we need to find the logarithm of 140 with base 3:
log₃(140) ≈ 4.64
So, the final answer is:
log₃(140) ≈ 4.64
**Step-by-Step Explanation:**
1. Write the logarithm in the desired form: log₃(140)
2. Use the change of base formula: log₃(x) = log(e) / log(e)/3
3. Evaluate the logarithm: log₃(140) ≈ 4.64
**b. log₂(27)**
To evaluate this logarithm, we can use the change of base formula:
log₂(x) = log(e) / log(e)/2
where e is the base of the natural logarithm.
log₂(27) = log(e) / log(2)
First, we need to find the logarithm of 27 with base 2:
log₂(27) ≈ 4.75
So, the final answer is:
log₂(27) ≈ 4.75
**Step-by-Step Explanation:**
1. Write the logarithm in the desired form: log₂(27)
2. Use the change of base formula: log₂(x) = log(e) / log(e)/2
3. Evaluate the logarithm: log₂(27) ≈ 4.75
**c. (log₃x)((logₓ9))**
To evaluate this expression, we need to follow the order of operations (PEMDAS):
1. Evaluate the inner expression: (logₓ9)
2. Evaluate the outer expression: (log₃x)((logₓ9))
**Step-by-Step Explanation:**
1. Evaluate the inner expression: (logₓ9)
We need to find the logarithm of 9 with base x:
(logₓ9) = y
Since x is unknown, we can't simplify further.
2. Evaluate the outer expression: (log₃x)((logₓ9))
Substitute the inner expression into the outer expression:
(log₃x)(y)
Now, we need to evaluate this expression. Since x and y are unknown, we can't simplify further.
So, the final answer is:
(log₃x)((logₓ9)) = ? (Unknown)
Note that this expression is not a well-defined mathematical expression without knowing the value of x or y.
I hope this helps! Let me know if you have any further questions or need clarification on any of these steps.
