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