Step-by-step explanation:
1a.
[tex] {a}^{n} = \frac{2}{5} [/tex]
We got to find
[tex]a {}^{ - 3n} [/tex]
First take that
[tex] {a}^{n} = \frac{2}{5} [/tex]
Take the log base a of both sides.
[tex] log_{a}(a {}^{n} ) = log_{a}( \frac{2}{5} ) [/tex]
Thus will cancel out the base a, and the log
[tex]n = log_{a}( \frac{2}{5} ) [/tex]
Multiply both sides by -3.
[tex] - 3n = - 3 log_{a}( \frac{2}{5} ) [/tex]
Apply the Power Rule.
[tex] - 3n = log_{a}(( \frac{2}{5}) {}^{ - 3} ) [/tex]
[tex] - 3n = log_{a}( \frac{ \frac{1}{8} }{ \frac{1}{125} } ) [/tex]
Which simplifies to
[tex] - 3n = log_{a}( \frac{125}{8} ) [/tex]
Rewrite this in expoent form.
[tex]a {}^{ - 3n} = \frac{125}{8} [/tex]
1b.
[tex]10 {}^{2m} = 16[/tex]
Find
[tex]10 {}^{ - m} [/tex]
Multiply both sides log x.
[tex] log(10 {}^{2m} ) = log(16) [/tex]
[tex]2m = log(16) [/tex]
Multiply both sides by -1/2.
[tex] - m = - \frac{1}{2} log(16) [/tex]
Apply Power Rule
[tex] - m = log(16 {}^{ - \frac{ 1}{2} } ) [/tex]
[tex] - m = log(2 {}^{4} {}^{ ( - \frac {1}{2}) } ) [/tex]
Apply Power of Power Rule
[tex] - m = log(2 {}^{ - 2} ) [/tex]
Which gives us
[tex] - m = log( \frac{1}{4} ) [/tex]
Rewrite this in expoent form.
[tex]10 {}^{ - m} = \frac{1}{4} [/tex]
