PLEASE HELP NOW!! Solve for x, rounding to the nearest integer if necessary. Fully justify your response by showing all algebraic steps. log_x(8)=log_5(2)

Question

PLEASE HELP NOW!!

Solve for x, rounding to the nearest integer if necessary. Fully justify your response by showing all algebraic steps.


`log_x(8)=log_5(2)`

Answer 1

`\textit{Logarithm inversion} \\\\ \cfrac{1}{\log_a b}\implies \cfrac{(1)/(1)}{(\log_c b)/(\log_c a)}\implies \cfrac{1}{1}\cdot \cfrac{\log_c a}{\log_c b}\implies \log_b a \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \log_x(8)=\log_5(2)\implies \log_x(2^3)=\log_5(2)\implies 3\log_x(2)=\log_5(2) \\\\\\ 3\cdot \cfrac{1}{\log_2(x)}=\cfrac{1}{\log_2(5)}\implies \cfrac{3}{\log_2(x)}=\cfrac{1}{\log_2(5)}\implies 3\log_2(5)=\log_2(x) \\\\\\ \log_2(5^3)=\log_2(x)\implies 5^3=x\implies 125=x`

bear in mind that logarithm inversion is derived from the change of base rule, make the logarithm a fraction, drop it to the denominator, invert away.