Laws of Logarithms

Without using a calculator use the change of base rule to evaluate

1. a) log(subscript8)32(hint use base 2)
=log(subscript2 32)/log(subscript2 8) = 5/3

b) log(subscript36)1296. This one doesnt even give you a hint, so how do you know which base to use? Does it have to be a factor of 36 and 1296? Cause for the first one I could have used base four instead of 2 then.

a) To evaluate log₈32 using the change of base rule, you can use base 2.

Step 1: Write the equation using the change of base rule:
log₈32 = log₂32 / log₂8

Step 2: Simplify the numerator:
log₂32 = log₂(2^5) = 5

Step 3: Simplify the denominator:
log₂8 = log₂(2^3) = 3

Step 4: Substitute the simplified values into the equation:
log₈32 = 5 / 3

b) To evaluate log₃₆1296, you need to find the appropriate base using the change of base rule.

Step 1: Choose a base that will simplify the problem. In this case, it doesn't have to be a factor of 36 or 1296, but it should be a base that you can work with easily.

Step 2: Write the equation using the change of base rule:
log₃₆1296 = logₐ1296 / logₐ36

Step 3: Since there are no hints given, you can choose any base that you prefer. In this case, let's choose base 2:
logₐ1296 / logₐ36 = log₂1296 / log₂36

Step 4: Simplify the numerator (log₂1296):
log₂1296 = log₂(2^7) = 7

Step 5: Simplify the denominator (log₂36):
log₂36 = log₂(2^2 * 3^2) = log₂(2^2) + log₂(3^2) = 2 + 2log₂3

Step 6: Substitute the simplified values into the equation:
log₃₆1296 = 7 / (2 + 2log₂3)

Please note that you can choose different bases for the logarithms, but the most commonly used bases are 10 (logarithms with no base written explicitly) and 2 (logarithms with subscripted 2).

log8(32) = log2(32)/log2(8) = 5/3

you are correct

log36(1296)
you don't need to change base to evaluate, since 1296 = 36^2. so log36(1296) = 2

However, now that you know that, you can use any root of 36 to change base:

log36(1296) = log6(1296)/log6(36) = 4/2 = 2