How do I Find The Domain, Vertical Asymptote, and X- intercept of this logarithmic function:

f(x)=6+log base 6 (x-3)

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To find the domain, vertical asymptote, and x-intercept of the given logarithmic function f(x) = 6 + log base 6 (x-3), we need to understand the properties and behavior of logarithmic functions.

1. Domain:
The domain of a logarithmic function consists of all positive real numbers. In this case, the function f(x) is defined for all values of (x-3) greater than 0, since the base of the logarithm is 6. Therefore, the domain of f(x) is (3, ∞).

2. Vertical Asymptote:
A vertical asymptote occurs when the value inside the logarithm becomes zero or undefined. In this case, since the logarithm has a base of 6, setting (x-3) = 0 gives x = 3 as the value, which will make the logarithm undefined. Therefore, the vertical asymptote is x = 3.

3. X-intercept:
An x-intercept is the value of x when the function f(x) crosses the x-axis, which means the value of f(x) is equal to zero. To find the x-intercept, set f(x) = 0 and solve for x:
6 + log base 6 (x-3) = 0
log base 6 (x-3) = -6

To remove the logarithm, we can rewrite the equation using exponential form:
6^(-6) = x-3
1/(6^6) = x-3
x = 1/(6^6) + 3

So, the x-intercept of the function f(x) is approximately x ≈ 3.00000185.

Remember, when dealing with logarithmic functions, it is important to consider the domain restrictions and the properties of logarithms.