How do you solve 13^(x-3) = 7^(-8x)?

2 Answers
Mar 5, 2016

x = (3log13)/ (log13+8log7) = 0.424374, nearly.

Explanation:

Equate logarithms ( of any base ) and solve the resulting linear equation in x.

Mar 5, 2016

color(red)(" A very detailed explanation!")

" "x=(3log(13))/(8log(91))" "->" "color(blue)(x=(3color(white)(.)root(91)(13))/8)

color(green)(~~0.2132" to 4 decimal places")

Explanation:

color(blue)("Principle used to solve this problem")

If you have log(a^b) then this can be written as blog(a)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given:" "13^(x-3)=7^(-8x)

color(brown)("Take logs of both sides:")

" "log(13^(x-3))" " =" "log (7^(-8x))

" "=> (x-3)log(13)" "=" "-8xlog(7)

color(brown)("Divide both sides by "log(13))

" "x-3" "=" "-8x xx(log(7))/(log(13)

color(brown)("Divide both sides by "-8x)

" "x/(-8x) -3/(-8x)" "=" "(log(7))/(log(13)

" "-1/8 +3/(8x)" "=" "(log(7))/(log(13)

color(brown)("Add "1/8" to both sides")

" "3/(8x)" "=" "(log(7))/(log(13)) + 1/8

color(brown)("Divide both sides by 3")

" "1/(8x)" " =" " (log(7))/(3log(13)) + 1/24

color(brown)("Multiply both sides by 8")

" "1/x" "=" "(8log(7))/(3log(13)) + 8/24

color(brown)("But "8/24 = 1/3 )

" " 1/x = (8log(7))/(3log(13)) + 1/3

" " 1/x=(8log(7)+log(13))/(3log(13))" "->"Corrected at this point"

color(brown)("Inverting everything")

" "x= (3log(13))/(8log(7)+log(13)

color(white)(.)

color(Red)("Corrected value for "x)
x~~ 3.3418/7.8747 = 0.4244 to 4 decimal places