If log x^3- log 10x = log 10^5 find x

We'll impose costraints of existence of
logarithms:

x^3>0 =>
x>0

10x>0 =>
x>0

We'll solve the equation adding log 10x both
sides:

log x^3 = log 10x + log
10^5

We'll apply the product rule of
logarithms:

log x^3 = log
x*10^6

Since the bases are matching, we'll apply one to one
rule:

x^3 = x*10^6

We'll
subtract 10^6x both sides:

x^3 - 10^6*x =
0

We'll factorize by x:

x(x^2
- 10^6) = 0

We'll set each factor as
zero:

x = 0

We'll reject this
answer since x>0.

x^2 - 10^6 =
0

We'll re-write the difference of
squares:

(x - 1000)(x + 1000) =
0

We'll set each factor as
zero;

x - 1000 = 0

x =
1000

x + 1000 = 0

x =
-1000

Since this answer is negative, we'll reject it,
too.

The only valid solution of the equation
is x = 1000.