Question

Solving an exponential equation by using logarithms: Exact answers in logarithmic form

Answer 1

`14^(-x+10)~~ = ~~13^(3x) \\\\[-0.35em] ~\dotfill\\\\ \log(14^(10-x))=\log(13^(3x))\implies (10-x)\log(14)=x\log(13^3) \\\\\\ 10-x=\cfrac{x\log(13^3)}{\log(14)}\implies \cfrac{10-x}{x}=\cfrac{\log(13^3)}{\log(14)}\implies \cfrac{10}{x}-\cfrac{x}{x}=\cfrac{\log(13^3)}{\log(14)}`

`\cfrac{10}{x}-1=\cfrac{\log(13^3)}{\log(14)}\implies \cfrac{10}{x}=\cfrac{\log(13^3)}{\log(14)}+1\implies \cfrac{10}{~~(\log(13^3))/(\log(14))+1~~}=x \\\\\\ \cfrac{10}{~~(\log(13^3)+\log(14))/(\log(14))~~}=x\implies \cfrac{10\log(14)}{\log(13^3)+\log(14)}=x`