If log ⁡ 10 5 + log ⁡ 10 ( 5 𝑥 + 1 ) = log ⁡ 10 ( 𝑥 + 5 ) + 1 log 10 ​ 5+log 10 ​ (5x+1)=log 10 ​ (x+5)+1, find 𝑥 x.

Given data & domain

• 5x+1 > 0, x+5 > 0 (log arguments must be positive).

Concept / Approach

• Use log properties: log ⁡ 𝑎 + log ⁡ 𝑏 = log ⁡ ( 𝑎 𝑏 ) loga+logb=log(ab) and 1 = log ⁡ 10 10 1=log 10 ​ 10.

Step-by-step calculation log ⁡ 10 [ 5 ( 5 𝑥 + 1 ) ] = log ⁡ 10 [ 10 ( 𝑥 + 5 ) ] log 10 ​ [5(5x+1)]=log 10 ​ [10(x+5)] ⇒ 25 𝑥 + 5 = 10 𝑥 + 50 ⇒25x+5=10x+50 15 𝑥 = 45 ⇒ 𝑥 = < 𝑚 𝑎 𝑟 𝑘 > 3 < / 𝑚 𝑎 𝑟 𝑘 > 15x=45⇒x=3

Verification5x+1 = 16 > 0, x+5 = 8 > 0 valid. Substitute to check equality.

Final Answer3