Logarithm question

Question

Log_1/x243=5

i couldn’t figure it out

i need to find the domain and the solution

Answer 1

Recall that y = log_bz if and only if b^y = z. (1)

Now b = 1/x in your example. 243 = z and y = 5.

If you plug this in, you get (1/x)⁵ = 243. (2)

Solve for x and you get x = 1/3, since, 243^1/5 = 3.

Domain: Once you have it in the form shown in equation (2), you can see the domain is all of ℜ

Proof of (1):

y = log_bx

y = log(x) / log(b) From the basic properties of logs.

y * log(b) = log(x)

log(b)^y = log(x)

e^log(b)y = e^log(x) take the exponent of both sides. the left side is exp(log(b)^y) but I can't write it.

b^y = x the exponent and log functions cancel out

Its a little tricky to make this work, but if you use the substitution 1/x = b, you can see it.

Answer 2

The equation given is equivalent to

(1/x)⁵ = 243 = 3⁵

which means1/x=3

You go from here.