$2(\log _3 x-1)=\log _3 x$ and $y=\sqrt{x}+1$

Solve the logarithmic equation 2 ( lo g 3 ​ x − 1 ) = lo g 3 ​ x for x , which gives x = 9 .
Substitute the value of x into the equation y = x ​ + 1 .
Calculate y = 9 ​ + 1 = 3 + 1 = 4 .
The final answer is 4 ​ .

Explanation

Problem Analysis We are given two equations:

2 ( lo g 3 ​ x − 1 ) = lo g 3 ​ x

y = x ​ + 1


Our goal is to find the value of y . To do this, we first need to solve the first equation for x , and then substitute that value into the second equation to find y .

Solving for x Let's solve the first equation for x :

2 ( lo g 3 ​ x − 1 ) = lo g 3 ​ x
Distribute the 2:
2 lo g 3 ​ x − 2 = lo g 3 ​ x
Subtract lo g 3 ​ x from both sides:
2 lo g 3 ​ x − lo g 3 ​ x = 2
lo g 3 ​ x = 2
Now, rewrite the equation in exponential form:
x = 3 2
x = 9

Solving for y Now that we have the value of x , we can substitute it into the second equation to find y :

y = x ​ + 1
Substitute x = 9 :
y = 9 ​ + 1
y = 3 + 1
y = 4

Final Answer Therefore, the value of y is 4.

Examples
Imagine you are designing a staircase where the height of each step ( y ) depends on the length of the base of the step ( x ) following the equation y = x ​ + 1 . If you also know that the length x is related to a logarithmic equation 2 ( lo g 3 ​ x − 1 ) = lo g 3 ​ x , solving these equations simultaneously helps you determine the exact height of each step. This ensures that the staircase is both aesthetically pleasing and ergonomically sound, providing a comfortable and safe ascent.

Answered by GinnyAnswer | 2025-07-04

We solved the logarithmic equation 2 ( lo g 3 ​ x − 1 ) = lo g 3 ​ x to find that x = 9 . Then, substituting this value into the equation y = x ​ + 1 gives us y = 4 .
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Answered by Anonymous | 2025-07-06