Answer :

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]