Solve for $$ x $$ $$ \log _{5}(3 x-7)=1 $$

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Solve the equation $$ \log_{5}(3x-7)=1 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{5}{\left(3x-7\right)}=1$$
- step1: Find the domain:
  $$\log_{5}{\left(3x-7\right)}=1,x>\frac{7}{3}$$
- step2: Convert the logarithm into exponential form:
  $$3x-7=5^{1}$$
- step3: Evaluate the power:
  $$3x-7=5$$
- step4: Move the constant to the right side:
  $$3x=5+7$$
- step5: Add the numbers:
  $$3x=12$$
- step6: Divide both sides:
  $$\frac{3x}{3}=\frac{12}{3}$$
- step7: Divide the numbers:
  $$x=4$$
- step8: Check if the solution is in the defined range:
  $$x=4,x>\frac{7}{3}$$
- step9: Find the intersection:
  $$x=4$$
The solution to the equation $$ \log_{5}(3x-7)=1 $$ is $$ x=4 $$.
$$ x=4 $$

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