Solve for x. Give exact answer. log(x + 4) - log x = 1

The logarithm base 10 of x is the exponent of 10 needed to produce x.
log10x = c means that where 10^c = x where x > 0
Logarithms base 10 are called common logarithms and log10x is written as log x.

log(x + 4) - log x = 1

log(x+4)-log(x) = 1

Use the quotient rule of logarithms:
log(a)-log(b) → log(a/b)

log(x+4)-log(x) = 1 (original equation)
log[(x+4)/x] = 1 (use quotient rule of logarithms)
10^[log(x+4)/x] = 10^(1) (exponentiate using base 10)
(x+4)/x = 10 (rewrite)
10x = x+4 (cross multiply)
9x = 4 (subtract x)
x = 4/9 (divide by 9)

Check:
log[(x+4)/x] = 1
log[(4/9+4)/(4/9)] = 1
log[(40/9)/(4/9)] = 1
log(10) = 1
1 = 1
True - the solution checks correctly

Solution: x = 4/9

ok im assuming u know the rules of logs so :

on the LHS :log(x + 4) - log x = log((x+4)/x)

so this now reads: log((x+4)/x) = 1
taking the exponent of 10 ( by the defination) we get (x+4)/x = 10^1

this means (x+4)/x = 10 (since 10^1 is 10)
multiply both sides by x : x+4 = 10x
subtract x from both sides : 4=10x-x
this is equal to 4=9x
so x = 4/9

u can check its true by plugging it into the original equation. hope this helped

if u dont know rules of logarithms its
for addition : log(a) + log(b) = log(ab)
for subtraction: log(a) - log(b) = log(a/b)

log(x + 4) - log x = 1
log[(x+4)/x] = 1
(x+4)/x = 10
x + 4 = 10x
4 = 9x
x = 9/4 = 2.25