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 2(5^x)=4^(x-1) what is the value of x?

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law222 | Student, Grade 11 | (Level 2) Honors

Posted January 8, 2013 at 1:18 PM via web

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 2(5^x)=4^(x-1) what is the value of x?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted January 8, 2013 at 1:30 PM (Answer #1)

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You need to take logarithms both sides, such that:

`ln(2*5^x) = ln (4^(x-1))`

You need to use the following logarithmic identities, such that:

`ln(a*b) = ln a + ln b`

`ln (a^b) = b* ln a`

Reasoning by analogy yields:

`ln(2*5^x) = ln (4^(x-1)) => ln 2 + ln 5^x = (x - 1) ln 4`

Since` ln 4 = ln 2^2 => ln 4 = 2 ln 2`

`ln 2 + x*ln 5 = 2(x - 1) ln 2`

Opening the brackets yields:

`ln 2 + x*ln 5 = 2xln 2 - 2ln 2`

Isolating the terms that contain x to the left side yields:

`x*ln 5 - 2xln 2 = -ln 2 - 2 ln 2`

Factoring out x yields:

`x(ln 5 - 2*ln 2) = -3 ln 2`

You need to multiply by -1 both sides such that:

`x(ln 4 - ln 5) = ln 2^3 => x*ln (4/5) = ln 8`

`x = (ln 8)/(ln(4/5))`

Hence, evaluating the solution to the given equation yields `x = (ln 8)/(ln(4/5))` .

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