To solve the equation [tex]\(\log_5(x+1) - \log_5(x-1) = 2\)[/tex], let's go through it step by step:
1. Understand the Logarithmic Properties:
The logarithmic property [tex]\( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \)[/tex] is applicable here. So, we apply this property:
[tex]\[
\log_5(x+1) - \log_5(x-1) = \log_5\left( \frac{x+1}{x-1} \right)
\][/tex]
2. Rewrite the Equation:
Now the equation becomes:
[tex]\[
\log_5\left( \frac{x+1}{x-1} \right) = 2
\][/tex]
3. Convert the Logarithmic Equation to Exponential Form:
The property of logarithms that we use here is [tex]\( \log_b(a) = c \iff b^c = a \)[/tex]. Applying this, we get:
[tex]\[
\frac{x+1}{x-1} = 5^2
\][/tex]
4. Simplify the Exponential Equation:
[tex]\(5^2 = 25\)[/tex], so the equation simplifies to:
[tex]\[
\frac{x+1}{x-1} = 25
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we clear the fraction by multiplying both sides by [tex]\(x-1\)[/tex]:
[tex]\[
x + 1 = 25(x - 1)
\][/tex]
Now, distribute the 25 on the right-hand side:
[tex]\[
x + 1 = 25x - 25
\][/tex]
Next, we'll collect all [tex]\(x\)[/tex] terms on one side and constant terms on the other side. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
1 = 24x - 25
\][/tex]
Now, add 25 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
26 = 24x
\][/tex]
Finally, divide by 24 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{26}{24} = \frac{13}{12}
\][/tex]
So, the solution to the equation [tex]\(\log_5(x+1) - \log_5(x-1) = 2\)[/tex] is:
[tex]\[
x = \frac{13}{12}
\][/tex]
