Given:
#color(red)(e^(2x-3) = 8)# ... [ 1 ]
We can use the formula:
If#" " f(x) = g(x),# then the following is also true:
#color(green)(ln [f(x) ] = ln [ g(x) ])#
Now, we can write [ 1 ] as
#color(red)(ln(e^(2x-3)) = ln(8))# ... [ 2 ]
#rArr ln(e^(2x-3)) = ln(8)#
We will consider the Left-Hand Side (LHS) first for simplification:
#rArr (2x-3) ln(e)#
We know that #color(red)(ln(e) = 1)#
Hence, we can conclude that
#rArr ln(e^(2x-3)) = (2x-3) * ln (e) = (2x-3)*1 = 2x-3# ...[ A ]
Again, from #color(red)(ln(e^(2x-3)) = ln(8))# ... [ 2 ],
we will consider the Right-Hand Side (RHS) next for simplification:
We have #color(red)(ln(8))#
#rArr ln(2^3)#
#rArr 3 ln(2)# .. [B]
Using [ A ] and [ B ], we get
#color(blue)(2x-3 = 3 ln 2)#
Add #3# to both sides:
We get,
#2x - 3 + 3 = 3 ln 2 + 3#
#rArr 2x - cancel 3 + cancel 3 = 3 ln 2 + 3#
#rArr 2x = 3 ln 2 + 3#
Divide by each term by #2# to further simplify:
#rArr (2x)/2 = (3 ln 2)/2 + 3/2#
#rArr (cancel 2x)/cancel 2 = (3 ln 2)/2 + 3/2#
#rArr x = [ 3 ln 2 + 3 ]/2#
If you want to solve using a calculator to obtain a numeric value,
then,
we use the following calculator results
#ln 2 = 0.693147#
#3 ln 2 = 2.079441542#
#3 ln 2 + 3 = 5.079441542#
# [ 3 ln 2 + 3 ]/2 =2.539720771#
Hence, our final solution is ...
#color(blue)(x = (3 ln(2) + 3)/2 " "color(green)[ or ] " "x ~~ 2.5397207)#
Hope you find this useful.
