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zepelin [54]
3 years ago
6

Given that 3^x = 4^y = 12^z, show that z = (xy)/(x+y).

Mathematics
1 answer:
irinina [24]3 years ago
8 0
3^{x} = 4^{y} = 12^{z}
3^{x} = 4^{y} = (4 \cdot 3)^{z}
3^{x} = 4^{y} = 4^{z} \cdot 3^{z}

\text{Let a } = 3^{x} = 4^{y} = 4^{z} \cdot 3^{z}
log_3a = x
log_4a = y
log_{(4 \cdot 3)}a = z

Using change of base:
x = \frac{lna}{ln3}
y = \frac{lna}{ln4}
z = \frac{lna}{ln(4 \cdot 3)}

ln3 = \frac{lna}{x}
ln4 = \frac{lna}{y}
ln(4 \cdot 3) = \frac{lna}{z}

Now, ln(4 · 3) = ln(4) + ln(3)

\frac{lna}{z} = \frac{lna}{x} + \frac{lna}{y}
\frac{1}{z} = \frac{1}{x} + \frac{1}{y}
\frac{1}{z} = \frac{x + y}{xy}

\therefore z = \frac{xy}{x + y}
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