Answer:
The solution for the given expression [tex]\log _{3x-2}\left(125\right)=3[/tex] is [tex]\frac{7}{3}[/tex]
Step-by-step explanation:
Given : Expression [tex]\log _{3x-2}\left(125\right)=3[/tex]
We have to find the solution for the given expression [tex]\log _{3x-2}\left(125\right)=3[/tex]
Consider the given expression [tex]\log _{3x-2}\left(125\right)=3[/tex]
Apply log rule, [tex]\log _a\left(b\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}[/tex]
[tex]\log _{3x-2}\left(125\right)=\frac{\ln \left(125\right)}{\ln \left(3x-2\right)}[/tex]
[tex]\frac{\ln \left(125\right)}{\ln \left(3x-2\right)}=3[/tex]
Multiply both side by [tex]\ln \left(3x-2\right)[/tex]
We get, [tex]\frac{\ln \left(125\right)}{\ln \left(3x-2\right)}\ln \left(3x-2\right)=3\ln \left(3x-2\right)[/tex]
Simplify , we have,
[tex]\ln \left(125\right)=3\ln \left(3x-2\right)[/tex]
Divide both sie by 3, we get,
[tex]\frac{3\ln \left(3x-2\right)}{3}=\frac{\ln \left(125\right)}{3}[/tex]
Also, [tex]\frac{\ln \left(125\right)}{3}=\frac{\ln \left(5^3\right)}{3}=\frac{3\ln \left(5\right)}{3}=\ln(5)[/tex]
Thus, [tex]\ln \left(3x-2\right)=\ln \left(5\right)[/tex]
When logs have same base, we have,
[tex]\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\quad \Rightarrow \quad f\left(x\right)=g\left(x\right)[/tex]
Thus, [tex]3x - 2 = 5[/tex]
Add 2 both sides, we have,
[tex]3x = 7[/tex]
Divide both side by 3, we have,
[tex]x=\frac{7}{3}[/tex]
Thus, the solution for the given expression [tex]\log _{3x-2}\left(125\right)=3[/tex] is [tex]\frac{7}{3}[/tex]
