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If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is?

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Re: If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is? [#permalink]
Lakshyamanchester wrote:
If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is?

A) 1/3
B) 1
C) 2/3
D) 2
E) 3

Explanation:

Given: (5.55)^x = (0.555)^y = 1000 =10^3

Taking log with respect to 10 ( that is base 10)

x log 5.55 = y log (0.555) = 3 log 10 = 3

x log 5.55 = 3
x = 3/log5.55
1/x = log 5.55/3 ---1

y log (0.555) = 3
y log (5.55/10) = 3
y log 5.55 – y log 10 = 3
y log 5.55 – y = 3
y = 3 /(log 5.55 -1)
1/y = (log 5.55 -1)/3 ---2

From equation 1 & 2

1/x -1/y = 1/3
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Re: If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is? [#permalink]
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Lakshyamanchester wrote:
If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is?

A) 1/3
B) 1
C) 2/3
D) 2
E) 3

Key concept: $$(b^n)^{\frac{1}{n}} = b$$

Take: $$5.55^x = 1000$$
Raise both sides to the power of $$\frac{1}{x}$$ to get: $$5.55 = 1000^{\frac{1}{x}}$$

Take: $$0.555^y = 1000$$
Raise both sides to the power of $$\frac{1}{y}$$ to get: $$0.555 = 1000^{\frac{1}{y}}$$

This means: $$\frac{5.55}{0.555}= \frac{1000^{\frac{1}{x}}}{1000^{\frac{1}{y}}}$$

Simplify both sides: $$10= 1000^{\frac{1}{x} - \frac{1}{y}}$$

Rewrite 1000 as follows: $$10= (10^3)^{\frac{1}{x} - \frac{1}{y}}$$

Simplify: $$10^1= 10^{3(\frac{1}{x} - \frac{1}{y})}$$

This means: $$1 = 3(\frac{1}{x} - \frac{1}{y})$$

Divide both sides by 3 to get: $$\frac{1}{3} = \frac{1}{x} - \frac{1}{y}$$

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Re: If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is? [#permalink]
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Re: If (5.55)^x = (0.555)^y = 1000, then the value of 1/x - 1/y is? [#permalink]
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