If `log_(k)A.log_(5)k=3,` then A=

Options

A. `5k^(3)`
B. k^(3)`
C. `125`
D. `243`

Answer

To solve the equation \( \log_k A \cdot \log_5 k = 3 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
Using the change of base formula for logarithms, we can express \( \log_k A \) and \( \log_5 k \) as:
\[ \log_k A = \frac{\log A}{\log k} \quad \text{and} \quad \log_5 k = \frac{\log k}{\log 5} \]

### Step 2: Substitute into the equation
Substituting these expressions into the original equation gives:
\[ \frac{\log A}{\log k} \cdot \frac{\log k}{\log 5} = 3 \]

### Step 3: Simplify the equation
Notice that \( \log k \) in the numerator and denominator cancels out:
\[ \frac{\log A}{\log 5} = 3 \]

### Step 4: Isolate \( \log A \)
Now, we can isolate \( \log A \) by multiplying both sides by \( \log 5 \):
\[ \log A = 3 \cdot \log 5 \]

### Step 5: Use the power property of logarithms
We can use the property of logarithms that states \( a \cdot \log b = \log(b^a) \):
\[ \log A = \log(5^3) \]

### Step 6: Remove the logarithm
Since the logarithms are equal, we can set the arguments equal to each other:
\[ A = 5^3 \]

### Step 7: Calculate \( A \)
Calculating \( 5^3 \):
\[ A = 125 \]

Thus, the value of \( A \) is \( 125 \).

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