Solve for x log base 8 of x+5- log base 8 of x-2=1

Problem

(log_8)(x+5)−(log_8)(x−2)=1

Solution

1. Apply the quotient rule for logarithms, which states that (log_b)(M)−(log_b)(N)=(log_b)(M/N)

(log_8)((x+5)/(x−2))=1

2. Rewrite in exponential form by using the definition (log_b)(y)=x⇔bx=y

(x+5)/(x−2)=8

3. Simplify the exponent on the right side of the equation.

(x+5)/(x−2)=8

4. Multiply both sides by the denominator (x−2) to clear the fraction.

x+5=8*(x−2)

5. Distribute the 8 into the parentheses.

x+5=8*x−16

6. Isolate the variable by subtracting x from both sides and adding 16 to both sides.

21=7*x

7. Solve for x by dividing both sides by 7.

x=3

8. Check the domain to ensure the arguments of the original logarithms are positive. Since 3+5>0 and 3−2>0 the solution is valid.

Final Answer

(log_8)(x+5)−(log_8)(x−2)=1⇒x=3

Want more problems? Check here!