Mathematics
Rockstar1234
2016-04-29 15:17:42
Given that 3^x = 4^y = 12^z, show that z = (xy)/(x+y).
ANSWERS
Gum275
2016-04-29 16:53:15

[latex]3^{x} = 4^{y} = 12^{z}[/latex] [latex]3^{x} = 4^{y} = (4 cdot 3)^{z}[/latex] [latex]3^{x} = 4^{y} = 4^{z} cdot 3^{z}[/latex] [latex] ext{Let a } = 3^{x} = 4^{y} = 4^{z} cdot 3^{z}[/latex] [latex]log_3a = x[/latex] [latex]log_4a = y[/latex] [latex]log_{(4 cdot 3)}a = z[/latex] Using change of base: [latex]x = frac{lna}{ln3}[/latex] [latex]y = frac{lna}{ln4}[/latex] [latex]z = frac{lna}{ln(4 cdot 3)}[/latex] [latex]ln3 = frac{lna}{x}[/latex] [latex]ln4 = frac{lna}{y}[/latex] [latex]ln(4 cdot 3) = frac{lna}{z}[/latex] Now, ln(4 · 3) = ln(4) + ln(3) [latex]frac{lna}{z} = frac{lna}{x} + frac{lna}{y}[/latex] [latex]frac{1}{z} = frac{1}{x} + frac{1}{y}[/latex] [latex]frac{1}{z} = frac{x + y}{xy}[/latex] [latex] herefore z = frac{xy}{x + y}[/latex]

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