Question

Solve for x:

`2 log x + 1/3 log 125x^3 = 3 log 2 + log 5`

Answer 1

`2 log x + 1/3 log(125x^3) = 3 log 2 + log 5`

Step 1: Simplify each logarithmic term:

⇒ 2 log x = log x²

⇒ `1/3 log (125x^3) = log(125x^3)^(1/3)`

= log(5 × x)

= log 5 + log x   ...`("Since"  125^(1/3) = 5)`

⇒ 3 log 2 = log 2³

= log 8

So the equation becomes:

log x² + log 5 + log x = log 8 + log 5

Step 2: Combine logarithmic terms:

Using the property log a + log b = log (ab):

log(x² × 5 × x) = log(8 × 5)

Simplify:

log(5x³) = log 40

Step 3: Equate the arguments of the logarithms:

Since log a = log b implies a = b, we get:

5x³ = 40

Step 4: Solve for x:

`x^3 = 40/5 = 8`

`x = root(3)(8)`

x = 2