Final answer:
To solve the logarithmic equation 6log₃(0.5x) = 1, you rewrite the log equation in exponential form and solve for x, which gives an approximate result of x ≈ 2.402.
Step-by-step explanation:
To solve the logarithmic equation 6log₃(0.5x) = 1, we first need to isolate the logarithm.
Let's divide both sides by 6 to get log₃(0.5x) = 1/6.
Next, we can rewrite the equation in exponential form:
3log₃(0.5x) = 31/6
Since the base of the logarithm and the base of the exponent are the same, they cancel each other out:
0.5x = 31/6
Now we just need to isolate x:
x = (31/6) / 0.5
Calculating 31/6 gives approximately 1.2009, and dividing by 0.5 gives:
x = 1.2009 / 0.5 = 2.4018
To three decimal places, the solution of the logarithmic equation is:
x ≈ 2.402