What is the exponent rule of logarithms?

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Answer 1 · 2016-09-16T20:30:46

${log}_{a} (m^{n}) = n {log}_{a} (m)$ $log_(a)(m^(n)) = n log_(a)(m)$

Consider the logarithmic number ${log}_{a} (m) = x$ $log_(a)(m) = x$ :

${log}_{a} (m) = x$ $log_(a)(m) = x$

Using the laws of logarithms:

$\Rightarrow m = a^{x}$ $=> m = a^(x)$

Let's raise both sides of the equation to $n$ $n$ th power:

$\Rightarrow m^{n} = {(a^{x})}^{n}$ $=> m^(n) = (a^(x))^(n)$

Using the laws of exponents:

$\Rightarrow m^{n} = a^{x n}$ $=> m^(n) = a^(xn)$

Let's separate $x n$ $xn$ from $a$ $a$ :

$\Rightarrow {log}_{a} (m^{n}) = x n$ $=> log_(a)(m^(n)) = xn$

Now, we know that ${log}_{a} (m) = x$ $log_(a)(m) = x$ .

Let's substitute this in for $x$ $x$ :

$\Rightarrow {log}_{a} (m^{n}) = n {log}_{a} (m)$ $=> log_(a)(m^(n)) = n log_(a)(m)$