Answer :
Certainly! Let's work through each part of the question step by step.
### Part a: Ama's Age and Her Sister's Age
1. Let the sister's current age be \( S \).
2. Ama's current age is given as two-thirds of her sister's age. So,
[tex]\[ \text{Ama's Age} = \frac{2}{3}S \][/tex]
3. Four years ago, the age of Ama and her sister can be stated as:
- Ama's age four years ago: \(\frac{2}{3}S - 4\)
- Sister's age four years ago: \(S - 4\)
4. It is given that four years ago, Ama's age was half of her sister's age. Therefore,
[tex]\[ \frac{2}{3}S - 4 = \frac{1}{2}(S - 4) \][/tex]
5. To solve for \( S \), we first clear the fraction by multiplying every term by 6 (the least common multiple of 2 and 3):
[tex]\[ 6\left(\frac{2}{3}S - 4\right) = 6\left(\frac{1}{2}(S - 4)\right) \][/tex]
6. This simplifies to:
[tex]\[ 4S - 24 = 3(S - 4) \][/tex]
7. Distribute on the right side:
[tex]\[ 4S - 24 = 3S - 12 \][/tex]
8. Now isolate \( S \):
[tex]\[ 4S - 3S = 24 - 12 \][/tex]
[tex]\[ S = 12 \][/tex]
9. Thus, the sister's current age is 12 years old.
### Part b: Solving the Logarithmic Equation
1. We start with the equation:
[tex]\[ \log(5x - 4) = \log(x + 1) + \log 4 \][/tex]
2. Using the properties of logarithms, specifically the addition property \(\log a + \log b = \log(ab)\), we can combine the logs on the right:
[tex]\[ \log(5x - 4) = \log(4(x + 1)) \][/tex]
3. If \(\log(a) = \log(b)\), then \(a = b\):
[tex]\[ 5x - 4 = 4(x + 1) \][/tex]
4. Expand the right side:
[tex]\[ 5x - 4 = 4x + 4 \][/tex]
5. Isolate \( x \):
[tex]\[ 5x - 4x = 4 + 4 \][/tex]
[tex]\[ x = 8 \][/tex]
6. Therefore, the value of \( x \) is 8.
In conclusion, the sister's current age is 12 years old, and the value of [tex]\( x \)[/tex] is 8.
### Part a: Ama's Age and Her Sister's Age
1. Let the sister's current age be \( S \).
2. Ama's current age is given as two-thirds of her sister's age. So,
[tex]\[ \text{Ama's Age} = \frac{2}{3}S \][/tex]
3. Four years ago, the age of Ama and her sister can be stated as:
- Ama's age four years ago: \(\frac{2}{3}S - 4\)
- Sister's age four years ago: \(S - 4\)
4. It is given that four years ago, Ama's age was half of her sister's age. Therefore,
[tex]\[ \frac{2}{3}S - 4 = \frac{1}{2}(S - 4) \][/tex]
5. To solve for \( S \), we first clear the fraction by multiplying every term by 6 (the least common multiple of 2 and 3):
[tex]\[ 6\left(\frac{2}{3}S - 4\right) = 6\left(\frac{1}{2}(S - 4)\right) \][/tex]
6. This simplifies to:
[tex]\[ 4S - 24 = 3(S - 4) \][/tex]
7. Distribute on the right side:
[tex]\[ 4S - 24 = 3S - 12 \][/tex]
8. Now isolate \( S \):
[tex]\[ 4S - 3S = 24 - 12 \][/tex]
[tex]\[ S = 12 \][/tex]
9. Thus, the sister's current age is 12 years old.
### Part b: Solving the Logarithmic Equation
1. We start with the equation:
[tex]\[ \log(5x - 4) = \log(x + 1) + \log 4 \][/tex]
2. Using the properties of logarithms, specifically the addition property \(\log a + \log b = \log(ab)\), we can combine the logs on the right:
[tex]\[ \log(5x - 4) = \log(4(x + 1)) \][/tex]
3. If \(\log(a) = \log(b)\), then \(a = b\):
[tex]\[ 5x - 4 = 4(x + 1) \][/tex]
4. Expand the right side:
[tex]\[ 5x - 4 = 4x + 4 \][/tex]
5. Isolate \( x \):
[tex]\[ 5x - 4x = 4 + 4 \][/tex]
[tex]\[ x = 8 \][/tex]
6. Therefore, the value of \( x \) is 8.
In conclusion, the sister's current age is 12 years old, and the value of [tex]\( x \)[/tex] is 8.
