Final answer:
. log_(3)(x+6)=(1)/(3)log_(3)8
x = 2
Step-by-step explanation:
To solve the logarithmic equation log₃(x+6) = (1/3) log₃ 8, we can use the property of logarithms that allows us to bring down exponents. Applying this property, the equation becomes log₃(x+6) = log₃ 8^(1/3). By matching the bases, we get x + 6 = 2, as 8^(1/3) equals 2. Solving for x, we find that x = 2. It's crucial to check for extraneous solutions by verifying that the original equation holds true for x = 2.
The given logarithmic equation is solved by equating the argument inside the logarithm to the exponent outside the logarithm, using the property of logarithms. This simplifies the equation to x + 6 = 2. Solving for x, we find that x = 2. However, it's essential to check for extraneous solutions by substituting x = 2 back into the original equation. When x = 2, both sides of the equation log₃(x+6) = (1/3) log₃ 8 simplify to the same value, confirming that x = 2 is a valid solution.
In conclusion, after solving the logarithmic equation, the final answer is x = 2. The solution is verified by substituting it back into the original equation, ensuring there are no extra