If log ab + log bc + log ac = 10 (for positive values of a, b and c), what is the value of log abc?

Question

Please explain the steps on how to figure this problem out.

Answer 1

Or just simply expand the multiplied logs using the product rule as you showed. Do this for each multiplied log and then simply add them up without dealing with exponents, etc.

 

Log ab = log a + log b

log bc = log b + log c

log ac = log a + log c

 

log ab + log bc + log ac = 2(log a + log b + log c)

2(log a + log b + log c) = 10

log a + log b + log c = 5

log(abc) = 5  (used inverse of product rule)

 

Answer 2

Note:

Log ab = log a + log b

Using this rule we can combine the logarithms:

 

log ab + log bc + log ac = 10

log[ab(bc)] + log ac = 10

log ab²c + log ac = 10

log [ab²c(ac)] = 10

log (a²b²c²) = 10

log(abc)² = 10

 

Note:  log aⁿ = n log a

 

2 log abc = 10

 

log abc = 5