4^x=256
Take the base-4 logarithm of both sides:
log 4 ( 4 x ) = log 4 ( 256 )
Using the laws of logarithms, the above equation can be simplified as:
x • log 4 ( 4) = log 4 ( 256 )
Since the base-4 logarithm of 4 is simply 1, the above equation can be rewritten
as:
x • (1) = log 4 ( 256 )
x = log 4 ( 256 )
This can be solved for x by using the change of base formula:
x=
log 256
log 4
x=4
Note: the logarithm used in this last calculation is the “normal” base-10
logarithm, usually designated with “log” on calculators. It is also possible to use
the natural logarithm function, which is usually marked as “ln” on calculators.
Both functions will give the same result.
2^(1+2x)=8
This problem is solved in the same manner, except that this time we need to use
the base-2 logarithm:
log 2 ( 21+2 x ) = log 2 8
Again using the laws of logarithms, this can be rewritten as:
(1+ 2x ) log 2 2 = log 2 8
And again, the base-2 logarithm of 2 is simply 1, which gives:
(1+ 2x ) (1) = log 2 8
1+ 2x = log 2 8
Using the base change formula again gives:
1+ 2x =
log8
log 2
1+ 2x = 3
Now solve for x:
1+ 2x −1 = 3−1
2x = 2
x=
2
2
x =1