Question

How do you solve `17^ { - 8x } = 4^ { - x - 5}`?

Answer 1

`x=0.0651472106`

Explanation:

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Whenever your variable is in the exponent you need to use logarithms to solve for it.

`17^(-8x)=4^(-x-5)`

`log17^(-8x)=log4^(-x-5)`

Using the rule of logs:

`-8xlog17=(-x-5)log4`

`-8xlog17=-xlog4-5log4`

`-8xlog17+xlog4=-5log4`

`8xlog17-xlog4=5log4`

`x(8log17-log4)=5log4`

`x=(5log4)/(8log17-log4)`

`x=(5(0.60205999))/(8(1.23044892)-0.60205999)=0.0651472106`

Answer 2

You need to isolate `x` to find a solution, so...

Explanation:

To get the variable down out of the exponent, use a log function.

`log 17^(-8x) = log 4^(-x-5)`

Exponent comes down: `(-8x) log17 = (-x-5) log 4`

`Log 4` is just a number, so let's multiply `log 4` into the parentheses to get this `x` by itself:

`-8x log 17 = -x log 4 - 5 log 4`

Add `xlog4` to both sides of the equation to get the `x's` together:

`-8x log 17 + x log 4= - 5 log 4`

Pull `x` out of both terms on the LHS (left-hand side):

`x(-8 log 17 + log 4) = - 5 log 4`

Divide both sides by: `-8 log 17 + log 4`:

`x = (- 5 log 4)/(-8 log 17 + log 4)`.

Each of the three terms on the RHS (right hand side) is just a number. Either leave this answer the way it is - in log form, or use a calculator and round to two or three places - according to your instructor's preferences.