If-4-x-5-y-20-z-what-is-z-in-term-of-x-and-y-

Question Number 101498 by bemath last updated on 03/Jul/20

If 4^{x} = 5^{y} = 20^{z} .

what is z in term of x and y .

★ ♠

Commented by bobhans last updated on 03/Jul/20

x \ln (4) = y \ln (5) = z (\ln (4) + \ln (5))

\Rightarrow (1) \ln (4) = \frac{y \ln (5)}{x}

\Rightarrow (2) z = \frac{y \ln (5)}{\ln (4) + \ln (5)} = \frac{y \ln (5)}{\frac{y \ln (5)}{x} + \ln (5)}

z = \frac{yx}{y + x} = \frac{x y}{x + y} ♡ ◊

Commented by Dwaipayan Shikari last updated on 03/Jul/20

4^{x} = 5^{y} = 20^{z} = k (k \neq 0)

4 = k^{\frac{1}{x}} 5 = k^{\frac{1}{y}} 20 = k^{\frac{1}{z}}

k^{\frac{1}{x}} k^{\frac{1}{y}} = k^{\frac{1}{z}}

\frac{1}{z} = \frac{1}{x} + \frac{1}{y} \Rightarrow z = {(\frac{1}{x} + \frac{1}{y})}^{- 1} = \frac{x y}{x + y} ★ L

Commented by bramlex last updated on 03/Jul/20

thank you both

Answered by RKT last updated on 03/Jul/20

4^{x} = 20^{z}

\Rightarrow 4^{x} = 4^{z} \times 5^{z}

\Rightarrow 4^{x - z} = 5^{z}

\Rightarrow 4^{\frac{x - z}{z}} = 5

\Rightarrow 4^{\frac{y}{z} (x - z)} = 5^{y}

\Rightarrow 4^{\frac{y}{z} (x - z)} = 4^{x} ∵ 4^{x} = 5^{y}

\Rightarrow \frac{y}{z} (x - z) = x

\Rightarrow x y - y z = z x

\Rightarrow x y = z x + y z

\Rightarrow x y = z (x + y)

\Rightarrow \frac{x y}{x + y} = z

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