Answer :
To solve the given problem, we need to rewrite logarithmic expressions in their equivalent exponential form. Let's go through each part step by step:
### Part (a)
Given: [tex]\(\log_2 32 = 5\)[/tex]
This logarithmic equation states that the power to which the base 2 must be raised to get 32 is 5. We can convert it to exponential form as follows:
1. Identify the base of the logarithm, which is 2.
2. Determine the exponent, which is given by the logarithm as 5.
3. Set the power (exponent) to the number: [tex]\( 2^5 = 32 \)[/tex].
Therefore, in the equation [tex]\(2^A = B\)[/tex]:
- [tex]\(A = 5\)[/tex], because that's the power (exponent) used in the conversion.
- [tex]\(B = 32\)[/tex], because that's the number we arrived at with the exponentiation.
### Part (b)
Given: [tex]\(\log_5 3125 = 5\)[/tex]
This logarithmic equation states that the power to which the base 5 must be raised to get 3125 is 5. We can convert it to exponential form as follows:
1. Identify the base of the logarithm, which is 5.
2. Determine the exponent, which is given by the logarithm as 5.
3. Set the power (exponent) to the number: [tex]\( 5^5 = 3125 \)[/tex].
Therefore, in the equation [tex]\(5^C = D\)[/tex]:
- [tex]\(C = 5\)[/tex], because that's the power (exponent) used in the conversion.
- [tex]\(D = 3125\)[/tex], because that's the number we arrived at with the exponentiation.
So the filled boxes are:
- [tex]\(A = 5\)[/tex] and [tex]\(B = 32\)[/tex] for part (a),
- [tex]\(C = 5\)[/tex] and [tex]\(D = 3125\)[/tex] for part (b).
### Part (a)
Given: [tex]\(\log_2 32 = 5\)[/tex]
This logarithmic equation states that the power to which the base 2 must be raised to get 32 is 5. We can convert it to exponential form as follows:
1. Identify the base of the logarithm, which is 2.
2. Determine the exponent, which is given by the logarithm as 5.
3. Set the power (exponent) to the number: [tex]\( 2^5 = 32 \)[/tex].
Therefore, in the equation [tex]\(2^A = B\)[/tex]:
- [tex]\(A = 5\)[/tex], because that's the power (exponent) used in the conversion.
- [tex]\(B = 32\)[/tex], because that's the number we arrived at with the exponentiation.
### Part (b)
Given: [tex]\(\log_5 3125 = 5\)[/tex]
This logarithmic equation states that the power to which the base 5 must be raised to get 3125 is 5. We can convert it to exponential form as follows:
1. Identify the base of the logarithm, which is 5.
2. Determine the exponent, which is given by the logarithm as 5.
3. Set the power (exponent) to the number: [tex]\( 5^5 = 3125 \)[/tex].
Therefore, in the equation [tex]\(5^C = D\)[/tex]:
- [tex]\(C = 5\)[/tex], because that's the power (exponent) used in the conversion.
- [tex]\(D = 3125\)[/tex], because that's the number we arrived at with the exponentiation.
So the filled boxes are:
- [tex]\(A = 5\)[/tex] and [tex]\(B = 32\)[/tex] for part (a),
- [tex]\(C = 5\)[/tex] and [tex]\(D = 3125\)[/tex] for part (b).
