i need help with logs. I don't understand them.

Here's 8

log_ee¹⁷ + log1000   =     ( remember log_aa^b  = b )     

17  + 3   =

20

Aren't they just trees that have been chopped up?

Logs are basically exponents.

Here's 7

4^2x+1 = 1 / 256^x+3     we can write the right hand side  as 256^-(x+3)

We can equate bases and solve for the exponents.....Note that 256  = 4⁴ ....so we have

4^2x+1 = [4⁴]^-(x+3)    and by a property of exponents, we have

4^2x+1 = 4^-4(x+3)

And solving for the exponents, we have

2x + 1 = -4(x+3)    simplify

2x + 1 = -4x - 12    add 4x to both sides and subtract 1 from both sides

6x = -13                divide both sides by 6

x = -13/6

Here's 9

log [ (³√x * y² ) / z ]      so we have

log ³√x + log y² - log z    =

log x ^1/3 + log y²  - log z  =

(1/3)log x + 2 log y - log z

Here's 10

logA = x   logB = y      ...so..... 3x + y   =

3logA + logB =

logA³ + logB =

log (A³ B)

Finally......here's 11

log₆₂₅ 5 + log₆ 216       we can use the change of base rule here ....  log_ab  =   log b / log a

So we have

(log5 / log 625)  + (log 216 / log 6) =

1/4  + 3  =

.25 + 3 =

3.25

And there you go....