How do you solve #8^x = 4 times 12^(2x)#?

1 Answer
Sep 16, 2016

#x = - (2 ln(2)) / (ln(18))#

Explanation:

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))#.