If log a + log b + log c = 0 then what is the value of abc?

Concept:

Formula of Logarithms:

\({a^b} = x\; \Leftrightarrow lo{g_a}x = b\), here a ≠ 1 and a > 0 and x be any number.

Properties of Logarithms:

1. \({\log _a}a = 1\)
2. \({\log _a}\left( {x.y} \right) = {\log _a}x + {\log _a}y\)
3. \({\log _a}\left( {\frac{x}{y}} \right) = {\log _a}x - {\log _a}y\)
4. \({\log _a}\left( {\frac{1}{x}} \right) = - {\log _a}x\)
5. \({\rm{lo}}{{\rm{g}}_a}{x^p} = p{\rm{lo}}{{\rm{g}}_a}x\)
6. \(lo{g_a}\left( x \right) = \frac{{lo{g_b}\left( x \right)}}{{lo{g_b}\left( a \right)}}\)

Calculation:

Given: log a + log b + log c = 0.

Using the product rule of logarithm, we get,

log a + log b + log c = log (ab) + log c = 0

log (abc) = 0

As we know that, log ⁡1 = 0

⇒ log (abc) = log 1

Cancelling the log on both sides we get,

⇒ abc = 1

Hence, the value of abc is 1.