Logs_1 Given that \(\ln(a)={@a@}\) , express: \[\log_{a}(x^{@n@})\] as a single multiple of a natural logarithm.

\(\log_{a}(x^{@ n @}) = \) [[input:ans1]] [[validation:ans1]]

]]> Below are the basic rules of logarithms.
\[\log_{b}(mn) = \log_{b}(m) +\log_{b}(n)\]
\[\log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) -\log_{b}(n)\]
\[\log_{b}(m^n) = n\cdot\log_{b}(m)\]

\[\log_{a}(m)=\frac{\log_{b}(m)}{\log_{b}(a)}\]

Hence \[\log_{a}(x^{@n@}) = {@n@}\log_{a}(x) = {@n@}\frac{\ln(x)}{\ln(a)}\]

\[ = {@n@}\frac{\ln(x)}{@a@}={@correctanswer@}\]

]]> 1 0 0 a:rand(8)+2; n:rand(8)+2; correctanswer:ev(n*ln(x)/a); wronganswer1:ev(n*a/ln(x)); wronganswer2:n*a*ln(x); [[feedback:Result]]

]]> When \(\displaystyle \ln(a)={@a@}, \log_{a}(x^{@n@}) ={@correctanswer@}\) 1 0 0 *10 dot 1 i cos-1 lang [ 0 {"author":[{"firstName":"","lastName":"","institution":"Unknown - Legacy question","year":"2025"}],"language":["en"],"isPartOf":"STACK Library","license":"cc-sa-4.0","additional":{"UoE":{"Notes":"This is automatically generated metadata."}},"freeform":{}} ans1 algebraic correctanswer 20 1 0 0 1 1 1 1 1 Result 1.0000000 1 1 0 AlgEquiv ans1 correctanswer 0 = 1.0000000 -1 Result-0-T = 0.0000000 1 Result-0-F 1 AlgEquiv ans1 wronganswer1 0 = 0.0000000 -1 Result-1-T Check the Log Laws

]]> = 0.0000000 2 Result-1-F 2 AlgEquiv ans1 wronganswer2 0 = 0.0000000 -1 Result-2-T Check the Log Laws

]]> = 0.0000000 -1 Result-2-F 1531295862 864648734 1099277693 1731809073 209765752 1273387083 1610313421 2071319725 73064170 1410745669 1363551006 1556989561 1814621817 1819508562 1 ans1 correctanswer Result 1.0000000 0.0000000 Result-0-T 2 ans1 wronganswer1 Result 0.0000000 0.0000000 Result-1-T 3 ans1 wronganswer2 Result 0.0000000 0.0000000 Result-2-T