OEFENING 4 Los op vir $$ x $$ : $$ 1 \quad 3^{x}-3^{x-2}=24 $$ $$ 2 \quad 3^{x+1}=3^{x}+\frac{2}{3} $$ $$ 3 \quad 3^{x}+3^{x+2}=10 $$ $$ 4 \quad 5^{x+1}-\frac{3}{25}=2.5^{x} $$ $$ 5 \quad 2^{x}+2^{x}+2^{x}=96 $$ $$ 6 \quad 12^{x}+12^{x}=288 $$ $$ 7 \quad 2^{x}+2^{x-2}=5 $$ $$ 8 \quad 2^{2 x}+4^{x+1}=80 $$

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Laten we elk van de oefeningen stap voor stap oplossen:

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**1. $$ 3^{x} - 3^{x-2} = 24 $$**

**Oplossing:**
$$
\begin{align*}
3^{x} - 3^{x-2} &= 24 \
3^{x-2} \cdot 3^{2} - 3^{x-2} &= 24 \quad (\text{Omdat } 3^{x} = 3^{x-2} \cdot 3^{2}) \
3^{x-2} (9 - 1) &= 24 \
3^{x-2} \cdot 8 &= 24 \
3^{x-2} &= 3 \quad (\text{delen door } 8) \
3^{x-2} &= 3^{1} \
x - 2 &= 1 \
x &= 3
\end{align*}
$$

**Antwoord:** $$ x = 3 $$

---

**2. $$ 3^{x+1} = 3^{x} + \frac{2}{3} $$**

**Oplossing:**
$$
\begin{align*}
3^{x+1} &= 3^{x} + \frac{2}{3} \
3 \cdot 3^{x} - 3^{x} &= \frac{2}{3} \quad (\text{Omdat } 3^{x+1} = 3 \cdot 3^{x}) \
2 \cdot 3^{x} &= \frac{2}{3} \
3^{x} &= \frac{1}{3} \
3^{x} &= 3^{-1} \
x &= -1
\end{align*}
$$

**Antwoord:** $$ x = -1 $$

---

**3. $$ 3^{x} + 3^{x+2} = 10 $$**

**Oplossing:**
$$
\begin{align*}
3^{x} + 3^{x+2} &= 10 \
3^{x} (1 + 9) &= 10 \quad (\text{Omdat } 3^{x+2} = 9 \cdot 3^{x}) \
3^{x} \cdot 10 &= 10 \
3^{x} &= 1 \
3^{x} &= 3^{0} \
x &= 0
\end{align*}
$$

**Antwoord:** $$ x = 0 $$

---

**4. $$ 5^{x+1} - \frac{3}{25} = 2 \cdot 5^{x} $$**

**Oplossing:**
$$
\begin{align*}
5^{x+1} - \frac{3}{25} &= 2 \cdot 5^{x} \
5 \cdot 5^{x} - 2 \cdot 5^{x} &= \frac{3}{25} \quad (\text{Omdat } 5^{x+1} = 5 \cdot 5^{x}) \
3 \cdot 5^{x} &= \frac{3}{25} \
5^{x} &= \frac{1}{25} \
5^{x} &= 5^{-2} \
x &= -2
\end{align*}
$$

**Antwoord:** $$ x = -2 $$

---

**5. $$ 2^{x} + 2^{x} + 2^{x} = 96 $$**

**Oplossing:**
$$
\begin{align*}
3 \cdot 2^{x} &= 96 \
2^{x} &= 32 \
2^{x} &= 2^{5} \
x &= 5
\end{align*}
$$

**Antwoord:** $$ x = 5 $$

---

**6. $$ 12^{x} + 12^{x} = 288 $$**

**Oplossing:**
$$
\begin{align*}
2 \cdot 12^{x} &= 288 \
12^{x} &= 144 \
12^{x} &= 12^{2} \
x &= 2
\end{align*}
$$

**Antwoord:** $$ x = 2 $$

---

**7. $$ 2^{x} + 2^{x-2} = 5 $$**

**Oplossing:**
$$
\begin{align*}
2^{x} + 2^{x-2} &= 5 \
2^{x-2} \cdot 4 + 2^{x-2} &= 5 \quad (\text{Omdat } 2^{x} = 4 \cdot 2^{x-2}) \
2^{x-2} (4 + 1) &= 5 \
2^{x-2} \cdot 5 &= 5 \
2^{x-2} &= 1 \
2^{x-2} &= 2^{0} \
x - 2 &= 0 \
x &= 2
\end{align*}
$$

**Antwoord:** $$ x = 2 $$

---

**8. $$ 2^{2x} + 4^{x+1} = 80 $$**

**Oplossing:**
$$
\begin{align*}
2^{2x} + 4^{x+1} &= 80 \
2^{2x} + (2^{2})^{x+1} &= 80 \quad (\text{Omdat } 4 = 2^{2}) \
2^{2x} + 2^{2x+2} &= 80 \
2^{2x} (1 + 4) &= 80 \quad (\text{Omdat } 2^{2} = 4) \
2^{2x} \cdot 5 &= 80 \
2^{2x} &= 16 \
2^{2x} &= 2^{4} \
2x &= 4 \
x &= 2
\end{align*}
$$

**Antwoord:** $$ x = 2 $$

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**Samenvatting van de oplossingen:**

1. $$ x = 3 $$
2. $$ x = -1 $$
3. $$ x = 0 $$
4. $$ x = -2 $$
5. $$ x = 5 $$
6. $$ x = 2 $$
7. $$ x = 2 $$
8. $$ x = 2 $$
Here are the solutions for each equation: 1. $$ x = 3 $$ 2. $$ x = -1 $$ 3. $$ x = 0 $$ 4. $$ x = -2 $$ 5. $$ x = 5 $$ 6. $$ x = 2 $$ 7. $$ x = 2 $$ 8. $$ x = 2 $$

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