1. Logarithms

2. Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7.e Each tick on the axes is one unit. Logarithms of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because a number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0.singularity

3. Definition The log of any number is the power to which the base must be raised to give that number. log(10) is 1 and log(100) is 2 (because 10 2 = 100). Example log 2 X = 8 2 8 = X X = 256

4. Example 1 10 log x = X “10 to the” is also the anti-log (opposite )

5. Log 23.5 = 1.371 Antilog 1.371 = 23.5 = 10 1.371

6. Logs used in Chem The most prominent example is the pH scale, but many formulas that we use require to work with log and ln. The pH of a solution is the -log([H+]), where square brackets mean concentration.

7. Example 2 Review Log rules log X = 0.25 Raise both side to the power of 10 (or calculating the antilog) 10 log x = 10 0.25 X = 1.78

8. Example 3 Review Log Rules Log c (a m ) = m log c (a) Solve for x 3 x = 1000 Log both sides to get rid of the exponent log 3 x = log 1000 x log 3 = log 1000 x = log 1000 / log 3 x = 6.29

9. Multiplying and Dividing logs log a x log b = log (a+b) log a/b = log (a-b) This holds true as long as the logs have the same base.

10. Problem 1 log (x) 2 – log 10 - 3 = 0

11. Try It Out Problem 1 Solution Solution

12. Problem 2 3.5 = ln 5x

13. Get rid of the ln by anti ln (e x ) e 3.5 = e ln 5x e 3.5 = 5x 33.1 = 5x 6.62 = x

14. Negative Logarithms We recall that 10 -1 means 1/10, or the decimal fraction, 0.1. What is the logarithm of 0.1? SOLUTION: 10 -1 = 0.1; log 0.1 = -1 Likewise 10 -2 = 0.01; log 0.01 = -2

15. Natural Logarithms The natural log of a number is the power to which e must be raised to equal the number. e =2.71828 natural log of 10 = 2.303 e 2.303 = 10 ln 10 = 2.303 e ln x = x

16. SUMMARY Common LogarithmNatural Logarithm log xy = log x + log yln xy = ln x + ln y log x/y = log x - log yln x/y = ln x - ln y log x y = y log xln x y = y ln x log x 1/y = (1/y )log x ln x 1/y =(1/y)ln x

17. In summary NumberExponential ExpressionLogarithm 100010 3 3 10010 2 2 1010 1 1 110 0 0 1/10 = 0.110 -1 1/100 = 0.0110 -2 -2 1/1000 = 0.00110 -3 -3

18. Simplify the following expression log 5 9 + log 2 3 + log 2 6 We need to convert to “Like bases” (just like fraction) so we can add Convert to base 10 using the “Change of base formula” (log 9 / log 5) + (log 3 / log 2) + (log 6 / log 2) Calculates out to be 5.535

19. ln vs. log? Many equations used in chemistry were derived using calculus, and these often involved natural logarithms. The relationship between ln x and log x is: ln x = 2.303 log x Why 2.303?

20. What’s with the 2.303; Let's use x = 10 and find out for ourselves. Rearranging, we have (ln 10)/(log 10) = number. We can easily calculate that ln 10 = 2.302585093... or 2.303 and log 10 = 1. So, substituting in we get 2.303 / 1 = 2.303. Voila!

21. Sig Figs and logs For a measured quantity, the number of digits after the decimal point equals the number of sig fig in the original number 23.5 measured quantity  3 sig fig Log 23.5 = 1.371 3 sig fig after the decimal point

22. More log sig fig examples log 2.7 x 10 -8 = -7.57 The number has 2 significant figures, but its log ends up with 3 significant figures. ln 3.95 x 10 6 = 15.189 the number has 5 3

23. OK – now how about the Chem. LOGS and Application to pH problems: pH = -log [H+] What is the pH of an aqueous solution when the concentration of hydrogen ion is 5.0 x 10 -4 M? pH = -log [H+] = -log (5.0 x 10 -4 ) = - (-3.30) pH = 3.30

24. Inverse logs and pH pH = -log [H+] What is the concentration of the hydrogen ion concentration in an aqueous solution with pH = 13.22? pH = -log [H+] = 13.22 log [H+] = -13.22 [H+] = inv log (-13.22) [H+] = 6.0 x 10 -14 M (2 sig. fig.)