For integer values of a and b, `b^(a)=8`. The ratio of a to b is equivalent to the ratio of c to d, where c and d are integers. What is the value of c when d=10?

Answer

To solve the problem step by step, we need to find integer values of \( a \) and \( b \) such that \( b^a = 8 \). Then, we will use the ratio of \( a \) to \( b \) to find \( c \) when \( d = 10 \).

### Step 1: Identify integer values of \( b \) and \( a \) such that \( b^a = 8 \)

We know that \( 8 \) can be expressed as \( 2^3 \). Therefore, we can set \( b = 2 \) and \( a = 3 \):
\[ b^a = 2^3 = 8 \]
Thus, we have:
- \( b = 2 \)
- \( a = 3 \)

### Step 2: Set up the ratio of \( a \) to \( b \)

Now, we can express the ratio of \( a \) to \( b \):
\[ \frac{a}{b} = \frac{3}{2} \]

### Step 3: Set up the ratio of \( c \) to \( d \)

According to the problem, the ratio of \( a \) to \( b \) is equivalent to the ratio of \( c \) to \( d \):
\[ \frac{a}{b} = \frac{c}{d} \]
Substituting the known values:
\[ \frac{3}{2} = \frac{c}{10} \]

### Step 4: Solve for \( c \)

To find \( c \), we can cross-multiply:
\[ 3 \cdot 10 = 2 \cdot c \]
This simplifies to:
\[ 30 = 2c \]
Now, divide both sides by 2:
\[ c = \frac{30}{2} = 15 \]

### Final Answer

Thus, the value of \( c \) when \( d = 10 \) is:
\[ \boxed{15} \]