What is the value of `log_(sqrt(b))"a" log_((c))"b" log_((a)^(1//2))` c ?

Options

A. 12
B. 24
C. `1//12`
D. `1//24`

Answer

To solve the expression \( \log_{\sqrt{b}} a \cdot \log_{c} b \cdot \log_{(a^{1/2})} c \), we can use the properties of logarithms. Let's break it down step by step.

### Step 1: Rewrite the logarithms using properties
We start by rewriting the logarithms using the change of base formula and the power rule of logarithms.

1. **Change of base formula**:
\[ \log_{x} y = \frac{\ln y}{\ln x} \]
2. **Power rule**:
\[ \log_{x} (y^n) = n \cdot \log_{x} y \]

Using these properties, we rewrite each term in the expression:

\[ \log_{\sqrt{b}} a = \frac{\ln a}{\ln \sqrt{b}} = \frac{\ln a}{\frac{1}{2} \ln b} = \frac{2 \ln a}{\ln b} \]

\[ \log_{c} b = \frac{\ln b}{\ln c} \]

\[ \log_{(a^{1/2})} c = \frac{\ln c}{\ln (a^{1/2})} = \frac{\ln c}{\frac{1}{2} \ln a} = \frac{2 \ln c}{\ln a} \]

### Step 2: Substitute back into the expression
Now we substitute these back into the original expression:

\[ \log_{\sqrt{b}} a \cdot \log_{c} b \cdot \log_{(a^{1/2})} c = \left(\frac{2 \ln a}{\ln b}\right) \cdot \left(\frac{\ln b}{\ln c}\right) \cdot \left(\frac{2 \ln c}{\ln a}\right) \]

### Step 3: Simplify the expression
Now we can simplify the expression:

\[ = \frac{2 \ln a}{\ln b} \cdot \frac{\ln b}{\ln c} \cdot \frac{2 \ln c}{\ln a} \]

Notice that \( \ln b \) in the numerator and denominator cancels out, and \( \ln a \) in the numerator and denominator also cancels out:

\[ = 2 \cdot 2 = 4 \]

### Final Answer
Thus, the value of the expression \( \log_{\sqrt{b}} a \cdot \log_{c} b \cdot \log_{(a^{1/2})} c \) is \( 4 \).

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