Find the value of a . if `log_(a)128 = 7`.

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the equation \( \log_a 128 = 7 \), we can follow these steps:

### Step 1: Rewrite the logarithmic equation in exponential form
The logarithmic equation \( \log_a 128 = 7 \) can be rewritten in its exponential form. This means that \( a^7 = 128 \).

**Hint:** Remember that the logarithmic form \( \log_b x = y \) can be rewritten as \( b^y = x \).

### Step 2: Express 128 as a power of 2
Next, we can express 128 as a power of 2. We know that:
\[ 128 = 2^7 \]

**Hint:** Try to break down the number into its prime factors to express it as a power.

### Step 3: Set the exponential equation
Now we can substitute \( 128 \) with \( 2^7 \) in our equation:
\[ a^7 = 2^7 \]

**Hint:** When two expressions are equal and both are raised to the same power, their bases must also be equal.

### Step 4: Equate the bases
Since the bases are the same, we can equate \( a \) to \( 2 \):
\[ a = 2 \]

**Hint:** If \( x^m = y^m \), then \( x = y \) if \( m \) is not zero.

### Conclusion
Thus, the value of \( a \) is \( 2 \).

**Final Answer:** \( a = 2 \)

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