Question

write the expand form of log a power m b power n/c power z​

Answer 1

The expanded form of the logarithm is m log a + n log b - z log c

Given logarithm,

$\boxed{ \large log ( \frac{ {a}^{m} \times {b}^{n} }{ {c}^{z } } ) }$

Quotient rule :

$log( \frac{p}{q} ) = log(p) - log(q)$

Product rule :

$log(p \times q) = log(p) + log(q)$

Power rule :

$log( {p}^{q} ) = q \times log(p)$

Using the above rules, We can try to expand the logarithm.

$= log ( \frac{ {a}^{m} \times {b}^{n} }{ {c}^{z } } )\\ \\ = log( {a}^{m} \times {b}^{n} ) - log( {c}^{z} ) \\ \\ = log( {a}^{m} ) + log( {b}^{n} ) - log( {c}^{z} ) \\ \\ = m \times log(a) + n \times log(b) - z \times log(c)$

Therefore, Expanded form of the logarithm is

m log a + n log b - z log c

Answer 2

$\Huge{\underline{\underline{\mathfrak{Answer \colon}}}}$

To expand the given expression

$\sf{ log( \frac{{a}^{m} \times {b}^{n} }{c {}^{z} } )}$

Quotient Rule,

$\boxed{\sf{log( \frac{x}{y} ) = log \: x \: - \: log \: y}}$

$\rightarrow \: \sf{log( {a}^{m} \times {b}^{n}) - log( {c}^{z} ) }$

Product Rule,

$\boxed{\sf{log(x.y) = log \: x \: + log \: y}}$

$\rightarrow \: \sf{log( {a}^{m}) + log( {b}^{n} ) - log( {c}^{z}) }$

Exponential Rule,

$\boxed{\sf{log( {x}^{y}) = y.log(x) }}$

$\large{\rightarrow \: \sf{m.log(a) + n.log(b) - z.log(c)}}$

Thus,

You will get m.log(a) + n.log(b) - z.log(c)