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# 7^a -7^a-5=117642√7

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7^a -7^a-5=117642√7

Find the value of a

Nov 6, 2017

#1
-1

There is no solution for a.   Nov 6, 2017
#2
+1

I  think this might be

7^a -7^(a-5)=117642√7

We  can write

7^a -  7^a / 7^5  = (2 * 3 * 2801) * 7 * 7^(1/2)

7^a -  7^a / 7^5  = (2 * 3 * 2801) *  7^(3/2)    multiply through  by  7^5

7^(a + 5)  - 7^a  = (2 * 3 * 2801) *  7^(13/2)      factor

7^a ( 7^5 - 1)  =   (2 * 3 * 2801) *  7^(13/2)

7^a ( 16806) =  ( 16806) * 7^(13/2)         divide  out  16806

7^a  =  7^(13/2)

a ⇒  13/2   Nov 6, 2017
#3
0

Solve for a:
7^a - 7^(a - 5) = 117642 sqrt(7)

Simplify and substitute x = -7^(a - 5).
7^a - 7^(a - 5) = -16806 (-7^(a - 5))
= -16806 x:
-16806 x = 117642 sqrt(7)

Divide both sides by -16806:
x = -7 sqrt(7)

Substitute back for x = -7^(a - 5):
-7^(a - 5) = -7 sqrt(7)

Multiply both sides by -1:
7^(a - 5) = 7 sqrt(7)

Take the logarithm base 7 of both sides:
a - 5 = (log(7 sqrt(7)))/(log(7))

Add 5 to both sides:
a = 5 + (log(7 sqrt(7)))/(log(7))

Log here is the natural "ln".

Nov 6, 2017
edited by Guest  Nov 7, 2017