Let's solve the given equation step-by-step:
[tex]\[
\log_8(x + 6) - \log_8(x) = \log_8(58)
\][/tex]
### Step 1: Apply Logarithm Properties
Start by using the properties of logarithms to combine the logarithmic terms on the left-hand side. Recall that [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex]. Applying this property, we get:
[tex]\[
\log_8\left(\frac{x + 6}{x}\right) = \log_8(58)
\][/tex]
### Step 2: Equate the Arguments
Since the logarithms have the same base and are equal, we can equate their arguments:
[tex]\[
\frac{x + 6}{x} = 58
\][/tex]
### Step 3: Solve the Rational Equation
Next, solve the rational equation for [tex]\(x\)[/tex]. Start by clearing the fraction:
[tex]\[
x + 6 = 58x
\][/tex]
### Step 4: Isolate [tex]\(x\)[/tex]
Rearrange the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
x + 6 - 58x = 0
\][/tex]
[tex]\[
6 - 57x = 0
\][/tex]
[tex]\[
6 = 57x
\][/tex]
[tex]\[
x = \frac{6}{57}
\][/tex]
### Step 5: Simplify the Fraction
Simplify the fraction [tex]\( \frac{6}{57} \)[/tex]:
Both 6 and 57 can be divided by their greatest common divisor (GCD), which is 3:
[tex]\[
x = \frac{6}{57} = \frac{6 \div 3}{57 \div 3} = \frac{2}{19}
\][/tex]
Thus, the solution to the equation is:
[tex]\[
x = \frac{2}{19}
\][/tex]
This is the reduced and final form of the solution.
