How do you solve #4^ { 8x } = 500#?

2 Answers
Oct 19, 2017

One can use any base logarithm that one likes on both sides:

#log_b(4^(8x)) = log_b(500)#

Use the property #log_b(A^C) = (C)log_b(A)#

#(8x)log_b(4) = log_b(500)#

Divide both sides by #8log_b(4)#:

#x = log_b(500)/(8log_b(4))#

Most calculators have base 10 and base e, therefore, I recommend that you use one of these two bases, for an exact representation of x:

#x = log_10(500)/(8log_10(4)) = ln(500)/(8ln(4))#

Here is an approximate number for x:

#x ~~ 0.560362#

Oct 19, 2017

#x = 0.56036#

Explanation:

#4^(8x) = 500#

Taking log on both sides,
#log4^(8x) = log 500#

#8x log 4 = log 500#

#8x = log 500/ log 4 = 2.699 / 0.6021 = 4.4829#

#x = 4.4829/8 ~~ 0.56036#