We have: #8^(x) = 4 times 12^(2 x)#
Let's apply the natural logarithm to both sides of the equation:
#=> ln(8^(x)) = ln(4 times 12^(2 x))#
Using the laws of logarithms:
#=> x ln(8) = ln(4) + 2 x ln(12)#
Let's express some numbers in terms of #2#:
#=> x ln(2^(3)) = ln(2^(2)) + 2 x ln(12)#
#=> 3 x ln(2) = 2 ln(2) + 2 x ln(12)#
#=> 3 x ln(2) - 2 x ln(12) = 2 ln(2)#
#=> x (3 ln (2) - 2 ln(12)) = 2 ln(2)#
#=> x (ln(2^(3)) - ln (12^(2))) = 2 ln(2)#
#=> x (ln((2^(3)) / (12^(2)))) = 2 ln(2)#
#=> x (ln((8) / (144))) = 2 ln(2)#
#=> x (ln ((1) / (18))) = 2 ln(2)#
#=> x (ln(1) - ln(18)) = 2 ln(2)#
#=> x (0 - ln (18)) = 2 ln(2)#
#=> - x ln(18) = 2 ln(2)#
#=> x ln(18) = - 2 ln(2)#
#=> x = - (2 ln(2)) / (ln(18))#
Therefore, the solution to the equation is #x = - (2 ln(2)) / (ln(18))#.
