Final answer:
To solve log₅(3x-6)=7, we rewrite it in exponential form to get 5· = 3x - 6, and then calculate 5· to find 3x-6 = 78125. Solving for x yields x ≈ 26043.6667, and rounding to the nearest thousandth gives x ≈ 26043.667.
Step-by-step explanation:
To solve the equation log₅(3x-6)=7, we first need to understand that the logarithm given is in base 5 and that it can be rewritten in exponential form. This means we are looking for the number which, when raised to the power of 7, gives us the result of 3x-6. Therefore, the equation becomes 5· = 3x - 6.
Let's find the value of 3x-6 by calculating 5·:
5· = 5· = 78125.
Once we have this value, we can substitute it back into the equation 3x - 6 = 78125, leading to 3x = 78131.
Dividing both sides of the equation by 3, we find x:
x = 78131 / 3x ≈ 26043.6667
To round this to the nearest thousandth, we look at the fourth decimal place which is a 6, so we round up the third decimal place to 7, giving us x ≈ 26043.667.