Find the value of `(b^2)^5log_bx`:

Options

A. `10^x`
B. `x^4`
C. 10x
D. `x^10`

Answer

To find the value of \((b^2)^5 \log_b x\), we can follow these steps:

### Step 1: Simplify the expression
We start with the expression:
\[ (b^2)^5 \log_b x \]
Using the power of a power property, we can simplify \((b^2)^5\) as follows:
\[ (b^2)^5 = b^{2 \cdot 5} = b^{10} \]
So, we rewrite the expression:
\[ b^{10} \log_b x \]

### Step 2: Apply the logarithmic property
Next, we can use the property of logarithms that states:
\[ a^m \log_a b = b^m \log_a a \]
In our case, we can apply this property to our expression:
\[ b^{10} \log_b x = x^{10} \log_b b \]

### Step 3: Simplify further
Since \(\log_b b = 1\), we can simplify our expression further:
\[ x^{10} \log_b b = x^{10} \cdot 1 = x^{10} \]

### Conclusion
Thus, the value of \((b^2)^5 \log_b x\) is:
\[ \boxed{x^{10}} \]

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