Given that \(109^{0.64} = x\), \(109^{0.83} = y\) and \(x^{z} = y^{5}\), then the value of \(z\) is close to:
6.48
Take logarithms (base 109). From \(109^{0.64} = x\), we have \(\log_{109} x = 0.64\); from \(109^{0.83} = y\), we have \(\log_{109} y = 0.83\).
From \(x^{z} = y^{5}\): \(z \log_{109} x = 5 \log_{109} y\).
Substitute: \(z \times 0.64 = 5 \times 0.83 = 4.15\).
Therefore \(z = \dfrac{4.15}{0.64} \approx 6.48\).
Hence, the value of \(z\) is close to 6.48.
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