how do you rewrite loga xy over m as a sum or difference of logs?

Question

can you help me rewrite log_a xy over m as a sum or difference of logs?

Answer 1

remember some of the properties of logarithms and exponents:

log_a(x) = b means that a^b = x

log_a(y) = c means that a^c = y

log_a(xy) = log_a(x) + log_a(y) = b + c

log_a(x/y) = log_a(x) - log_a(y) = b - c

using a numerical example with a = 2, x = 4, and y = 8

log₂ (4*8) = log₂(4) + log₂(8) = 2 + 3 = 5

you can check that since 2⁵ = 32 = 8*4

 

again using the same numbers for a, x, and y but this time dividing

log₂(4/8) = log₂(4) - log₂(8) = 2 - 3 = -1

since 2^-1 = 1/2 = 4/8

Now, you just combine the two steps

this time I'll add an "m" value for you but will leave the writing of the general equation to you:

for a = 2, x = 4, y = 8, m = 32

log₂[(4*8)/32] = log₂(4) + log₂(8) - log₂32 = 2 + 3 - 5 = 0

2⁰ = (32/32) = 1