SOLUTION: Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph: y = log3(x-2) + 4 Please show me how you got the answers, thank you.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph: y = log3(x-2) + 4 Please show me how you got the answers, thank you.      Log On


   

Question 1130365: Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph: y = log3(x-2) + 4
Please show me how you got the answers, thank you.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
if you have this:
y+=+log%283%2C%28x-2%29%29+%2B+4+
domain - The domain of y is the set of all x values such that x+-+2+%3E+0+
Solve the above inequality to obtain the domain: x+%3E++2+
{ x element R: x+%3E++2+ }
The range of y is given by the interval:
(-+infinity+, +infinity).

b.
The vertical asymptote is obtained by solving the equation: x+-+2+=+0+
which gives x+=++2+
As x+approaches 2 from the right (x+%3E+2) , y decreases without bound because there is a vertical asymptote. How do we know this?


c.
- To find the x - intercept we need to solve the equation y+=+0 or
log%283%2C%28x+-2%29%29%2B4+=+0
log%283%2C+%28x+-2%29%29+=+-4
Rewrite the above equation in exponential form:

x+-+2+=+3%5E-4
x+-+2+=+1%2F3%5E4
x+-+2+=+1%2F81
x++=+1%2F81%2B2
x++=+1%2F81%2B162%2F81
x++=+163%2F81
The x intercept is at the point
The y intercept is at the point (0 , y%280%29)
=> (0 , log%283%2C+%280+-2%29%29%2B4) =>log%283%2C+%280+-2%29%29%2B4 have complex solution, there is no y intercept
d - So far we have the domain, range, x and y+intercepts and the vertical asymptote. We need more points to draw a graph.

table:
163%2F81 | 0
2.5 | 3.4
4 | 4.6
5 | 5
6 | 5.3
8 | 5.6
10 | 5.89
plot points and draw a graph: