Question

3log5(x-10)-1/2log54=5log52

Solve each equation. #10

Answer 1

`\large\boxed{x=14}`

Explanation:

`3\log_5(x-10)-(1)/(2)\log_54=5\log_52\\\\\text{Domain:}\ x-10>0\to x>10\\\\\text{use}\\\\\log_ab^n=n\log_ab\\\\\log_ab-\log_ac=\log_a\left((b)/(c)\right)\\========================\\\\\log_5(x-10)^3-\log_54^(1)/(2)=\log_52^5\qquad\text{use}\ a^(1)/(2)=√(a)\\\\\log_5(x-10)^3-\log_5\sqrt4=\log_532\\\\\log_5(x-10)^3-\log_52=\log_532\\\\\log_5((x-10)^3)/(2)=\log_532\iff((x-10)^3)/(2)=32\qquad\text{multiply both sides by 2}\\\\(x-10)^3=64\to x-10=\sqrt[3]{64}\\\\x-10=4\qquad\text{add 10 to both sides}\\\\x=14\in D`