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

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 y b 10 e

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 • 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 • 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 b b 2 2 2 b b b 2 2 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 Graph Logarithm Graph

6. It is simple to graph exponentials. For instance, to graph y = 2x, 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 20 = 1, then log (1) = 0, so (1, 0) is on the graph - Since 21 = 2, then log (2) = 1, so (2, 1) is on the graph - Since 22 = 4, then log (4) = 2, so (4, 2) is on the graph - Since 23 = 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