Question

solve the equation.
\log_{5}(x + 3) - \log_{5}x = 2

select the correct choice below and fill in any answer boxes present in

a. \\ x = \square
(simplify your answer, including any radicals. use integers or fra
b. \\ there is no solution.

Explanation:

Step1: Apply log quotient rule

Using the property $\log_a M - \log_a N = \log_a \frac{M}{N}$, we rewrite the left - hand side of the equation $\log_5(x + 3)-\log_5x = 2$ as $\log_5\frac{x + 3}{x}=2$.

Step2: Convert logarithmic to exponential form

Recall that if $\log_a b=c$, then $b = a^c$. For the equation $\log_5\frac{x + 3}{x}=2$, we have $\frac{x + 3}{x}=5^2$. Since $5^2 = 25$, the equation becomes $\frac{x + 3}{x}=25$.

Step3: Solve the rational equation

Multiply both sides of the equation $\frac{x + 3}{x}=25$ by $x$ (assuming $x
eq0$, and we will check the solution later) to get $x + 3=25x$.
Subtract $x$ from both sides: $3=25x - x$.
Simplify the right - hand side: $3 = 24x$.
Divide both sides by 24: $x=\frac{3}{24}=\frac{1}{8}$.

Step4: Check the domain

For the original logarithmic equation $\log_5(x + 3)-\log_5x = 2$, the arguments of the logarithms must be positive. So we need $x+3>0$ and $x > 0$.
If $x=\frac{1}{8}$, then $x+3=\frac{1}{8}+3=\frac{1 + 24}{8}=\frac{25}{8}>0$ and $x=\frac{1}{8}>0$. So $x=\frac{1}{8}$ is in the domain.

Answer:

$\frac{1}{8}$