Answer
See the explanation
Work Step by Step
Solving for $\log_{3}(x-1)=4,$ we use expoential form to solve logarithmic equations. Thus, we follow the following steps
1. Expressing the equation in the form $\log_{b}M=c,$
2. Use the definition of a logarithm to rewrite the exponential form, $\log_{b}M=c,$ means $b^c=M.$
3. Solve for the variable
4. Check proposed solutions in the original equation. Include in the solution set only values for which $M>0.$
Therefore, $\log_{3}(x-1)=4,$
$x-1=3^4,$
$x=3^4+1,$
$x=82.$
And solving for $\log_{3}(x-1)=\log_{3}4.$ We use the one-to-one property of logarithms to solve for exponential equations.
1. Expressing the equation in the form, $\log_{b}M=\log_{b}N.$
This form involves a single logarithms whose coefficient is $1$ on each side of the equation.
2. Use the one-to-one property to rewrite the equation with out logarithms: If $\log_{b}M=\log_{b}N,$ then $M=N.$
3. Solve for the variable.
4. Check the proposed solutions in the original equation. Include in the solutions set only values for which $M>0$ and $N>0.$
$\log_{3}(x-1)=\log_{3}4,$
$x-1=4,$
$x=5$
