# Calculate in logarithms of bases 2,5,25: L = (log base 2 (4) + log base 5 (1/5) - log base 25 (625)) ^225?

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Expert Answers

sciencesolve | Certified Educator

You need to use the logarithmic identities, such that:

`log_a b^n = n*log_a b`

`log (a/b) = log a - log b`

`log_a a = 1`

`log_a 1 = 0`

Reasoning by analogy, yields:

`log_2 4 = log_2 (2^2) => log_2 4 = 2log_2 2 = 2`

`log_25 625 = log_25 (25^2) => log_25 625 = 2 log_25 25 = 2`

`log_5(1/5) = log_5 1 - log_5 5 => log_5 (1/5) = -1`

Replacing 2 for `log_2 4, log_25 625` and -1 for` log_5(1/5)` , yields:

`L = (2 - 1 - 2)^225`

Reducing duplicate members yields:

`L = (-1)^225 => L = -1`

**Hence, evaluating the value of L, using the logarithmic identities, yields **`L = -1.`