Question

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

Answer 1

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`

Answer 2

`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`