Question

solve the equation. log _7(x + 1) - log _7x = 2 select the correct choice below and fill in any answer box a. x = (simplify your answer, including any radicals. use b. there is no solution.

Explanation:

Step1: Use log - subtraction rule

According to the rule $\log_a M-\log_a N=\log_a\frac{M}{N}$, we can rewrite the left - hand side of the equation. So, $\log_7(x + 1)-\log_7x=\log_7\frac{x + 1}{x}$. The equation becomes $\log_7\frac{x + 1}{x}=2$.

Step2: Convert to exponential form

By the definition of logarithms, if $\log_a b=c$, then $b = a^c$. Here, $a = 7$, $b=\frac{x + 1}{x}$, and $c = 2$. So, $\frac{x + 1}{x}=7^2=49$.

Step3: Solve for x

Multiply both sides of the equation $\frac{x + 1}{x}=49$ by $x$ (assuming $x
eq0$) to get $x + 1=49x$. Then subtract $x$ from both sides: $1=49x−x$. Combining like terms gives $1 = 48x$. Divide both sides by 48, so $x=\frac{1}{48}$.

Answer:

A. $x=\frac{1}{48}$