A logarithmic function of the form f(x) = log_b x+c has an x-intercept at x = 1 and a vertical asymptote at y = 0. Also, f(x) intersects with the line y=1 when x=8.
what is f(x)
plug in your numbers.
log_b(1) = 0, so c=0
log_b(8) = 1, so b=8
y = log8x
To find the logarithmic function f(x), we can start by using the given information about the x-intercept and vertical asymptote.
1) X-intercept at x = 1:
An x-intercept occurs when the value of f(x) is equal to zero. Since the x-intercept is at x = 1, we can substitute this value into the logarithmic function: f(1) = log_b 1 + c = 0. Since log_b 1 = 0 for any base b, we can simplify the equation to c = 0.
2) Vertical asymptote at y = 0:
A vertical asymptote occurs when x approaches a certain value, and the function approaches positive or negative infinity. In this case, the vertical asymptote is at y = 0, which means that as x approaches a certain value, f(x) approaches negative infinity. A logarithmic function approaches negative infinity as x approaches zero. Therefore, the base of the logarithmic function must be greater than 1.
3) Intersection with the line y = 1 at x = 8:
We are given that f(x) intersects with the line y = 1 when x = 8. Substituting these values into the logarithmic function, we get: f(8) = log_b 8 + c = 1.
To simplify further, we can rewrite log_b 8 as log_b (2^3), which is equivalent to 3 log_b 2, using the logarithmic property. Therefore, we have: 3 log_b 2 + c = 1.
Combining all the obtained information, we have the following equations:
1) c = 0
2) 3 log_b 2 + c = 1
From equation 1, we know that c = 0. Substituting this value into equation 2, we get: 3 log_b 2 + 0 = 1.
Rearranging the equation, we have: 3 log_b 2 = 1. Divide both sides by 3: log_b 2 = 1/3.
To find the value of b, we need to rewrite the equation in exponential form. In logarithms, log_b x = y is equivalent to b^y = x. Applying this property to the equation, we get: b^(1/3) = 2.
Cubing both sides of the equation, we have: b = 2^3 = 8.
Therefore, the logarithmic function f(x) is given by: f(x) = log_8 x