4. Let $a, b, c$ and $d$ be real numbers greater than 1. Calculate the value of the expression

$$
a^{\left(\log _{b} c\right)-1} b^{\left(\log _{c} d\right)-1} c^{\left(\log _{d} a\right)-1} d^{\left(\log _{a} b\right)-1}
$$

if you know that $\log _{b} a \cdot \log _{d} c=1$
III/4. Let $S=a^{\left(\log _{b} c\right)-1} b^{\left(\log _{c} d\right)-1} c^{\left(\log _{d} a\right)-1} d^{\left(\log _{a} b\right)-1}$. We write $1=\log _{b} a \cdot \log _{d} c=$ $\frac{\log a \cdot \log c}{\log b \cdot \log d}$. Using this, we calculate

$$
\begin{aligned}
& \log (a b c d S)=\log \left(a^{\log _{b} c} b^{\log _{c} d} c^{\log _{d} a} d^{\log _{a} b}\right)= \\
& =\log a^{\log _{b} c}+\log b^{\log _{c} d}+\log c^{\log _{d} a}+\log d^{\log _{a} b}= \\
& =\log _{b} c \cdot \log a+\log _{c} d \cdot \log b+\log _{d} a \cdot \log c+\log _{a} b \cdot \log d= \\
& =\frac{\log c \cdot \log a}{\log b}+\frac{\log d \cdot \log b}{\log c}+\frac{\log a \cdot \log c}{\log d}+\frac{\log b \cdot \log d}{\log a}= \\
& =\frac{\log a \cdot \log c \cdot(\log b+\log d)}{\log b \cdot \log d}+\frac{\log b \cdot \log d \cdot(\log a+\log c)}{\log a \cdot \log c}= \\
& =\log a+\log b+\log c+\log d=\log (a b c d)
\end{aligned}
$$

from which it follows that $a b c d S=a b c d$ or $S=1$.

![](https://cdn.mathpix.com/cropped/2024_06_07_a8720c28cfa2c07912d2g-12.jpg?height=211&width=1650&top_left_y=908&top_left_x=206)