`log_3(18x+7)=log_3(3x+38)` Solve the equation. Check for extraneous solutions.

To solve the equation `log_3(18x+7)=log_3(3x+38)` , we apply logarithm property: `a^(log_a(x))=x` .

Raised both sides by base of `3` .

`3^(log_3(18x+7))=3^(log_3(3x+38))`

`18x+7=3x+38`

Subtract 7 from both sides.

`18x+7-7=3x+38-7`

`18x=3x+31`

Subtract 3x from both sides.

`18x-3x=3x+31-3x`

`15x=31`

Divide both sides by `15` .

`(15x)/15=31/15`

`x=31/15`

Checking: Plug-in `x=31/15` on `log_3(18x+7)=log_3(3x+38).`

`log_3(18*31/15+7)=?log_3(3*31/15+38)`

`log_3(186/5+7)=?log_3(31/5+38)`

`log_3(186/5+35/5)=?log_3(31/5+190/5)`

`log_3(221/5)~~log_3(221/5)` TRUE

or `3.449 ~~3.449.` TRUE

Thus, there is no extraneous solution. The `x=31/15` is a real solution for the given equation `log_3(18x+7)=log_3(3x+38)` .

Comments

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