Question

solve the equation.
log₇(x + 2) - log₇x = 2

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

a. x =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.
b. there is no solution.

Explanation:

Step1: Apply log - subtraction rule

Using the rule $\log_aM-\log_aN = \log_a\frac{M}{N}$, we rewrite the left - hand side of the equation: $\log_7\frac{x + 2}{x}=2$.

Step2: Convert to exponential form

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

Step3: Solve for x

Multiply both sides by $x$: $x + 2=49x$. Then subtract $x$ from both sides: $2=49x−x = 48x$. Divide both sides by 48: $x=\frac{2}{48}=\frac{1}{24}$.

Answer:

A. $x=\frac{1}{24}$