To write an equation for a logarithm graph, we start with the general form y equals a log base b of x minus h plus k. This represents transformations of the basic logarithm function. The blue curve shows the basic logarithm y equals log x, while the red curve shows a transformed version with vertical stretch, horizontal shift, and vertical shift.
Step one is to identify the vertical asymptote. Look for the vertical line that the logarithm graph approaches but never crosses. In this example, the vertical asymptote is at x equals 2, which means our horizontal shift h equals 2. This asymptote tells us that the argument of the logarithm is x minus 2.
Step two is finding the vertical shift k. Look for the point where x equals h plus 1. Since h equals 2, we look at x equals 3. At this point, the graph has y-coordinate 1, so k equals 1. This means our logarithm function is shifted up by 1 unit from the basic logarithm.
Step three is finding the base and stretch factor. We use another clear point on the graph. At point (12, 3), we substitute into our equation: 3 equals a log of 12 minus 2 plus 1. This gives us 2 equals a log of 10. Since log base 10 of 10 equals 1, we get a equals 2. So our stretch factor is 2.
Now we can write the final equation. Combining all our findings: h equals 2, k equals 1, a equals 2, and base 10, we get y equals 2 log of x minus 2 plus 1. This equation represents a logarithm with horizontal shift right 2 units, vertical shift up 1 unit, and vertical stretch by factor 2. The graph confirms our equation passes through the key points we identified.