Question

Solve the logarithmic equation: x= log5 (1/625)=

Answer 1

You can convert (1/625) to an exponent, and it would be ideal to have 5 as the base of it because you want your log base to cancel it out. what i usually do in this case is just test out 5^1, 5^2, etc until i find one that matches the number i need. in this case because the number you're trying to work with is a small fraction, you'll want to use NEGATIVE exponents so it'll create a fraction instead of a large whole number:

5^-1 = 1/5
. . . keep trying those. . .
5^-4 = 1/625

so, because they're equal to one another, it'll be waaay easier after you substitute 5^-4 in place of 1/625

x = log₅ 5⁻⁴

log base 5 of 5 simplifies to 1. subbing in the 5^-4 gets rid of the log for you altogether, and your -4 exponent drops down:

x = -4 is your answer

if the exponent dropping down doesn't make sense to you, you can think of it in another way:

x = log₅ 5⁻⁴

expand the expression so that the exponent moves in front of the log function:

x = (-4) log₅ 5

then, still, log base 5 of 5 simplifies to 1, so you're left with:

x = (-4)1 or x = -4