1. Logarithms and Their Graphs By: Jesus Rocha Period 2Pre-Calculus  John Napier (creator of logarithms)

2. Base b in Logarithm Problems The logarithm to the base b of x, log x, is the power to which you need to raise b in order to get x. log x = y means b = x Logarithmic Form Exponential form Rules: 1. Log x is only defined if b and x are both positive, and b ≠1 2. Log x is called the common logarithm of x, and is sometimes written as log 10. 3. Log x is called the natural logarithm of x b b b 10 e y

3. Solving Logarithms If log 1,000 = 3 (or the logarithm to the base 10 of 1,000 is 3) then its exponential form would be 10 = 1,000 Solving: - Move base 10 to the left of log (10 log 1,000 = 3) - It is easy to figure out that 10 to the power of 3 equals 1,000 so the exponential form would be written as 10 = 1,000 10 3 3

4. Laws of Logarithms 1.If the logs are being asked to be multiplied, log x (mn), then you should add the Logs: log m + log n ex: log (4x8) = log (4) + log (8) = 2+3=5 2.If the logs are being asked to be divided, log (m/n), then you should subtract the Logs: log m – log n ex: log (8/4) = log (8) – log (4) = 3-2=1 3. b = 1 b bb 222 b bb 22 2 0

5. Graphing Logarithms By nature of the logarithms, most log graphs tend to have the same shape, looking similar to a square root graph: Square Root GraphLogarithm Graph

6. It is simple to graph exponentials. For instance, to graph y = 2 x, you would just plug in some values for x, compute the corresponding y- values, and plot the points. A negative number or 0 would make it a little more difficult to solve: - Since 2 0 = 1, then log (1) = 0, so (1, 0) is on the graph - Since 2 1 = 2, then log (2) = 1, so (2, 1) is on the graph - Since 2 2 = 4, then log (4) = 2, so (4, 2) is on the graph - Since 2 3 = 8, then log (8) = 3, so (8, 3) is on the graph - Since 3, 5, 6, and 7 aren’t powers of 2, they wouldn’t work well with each other 2 2 2 2

7. The Results