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Logarithms

Logarithms. Logarithms. Logarithms to various bases: red is to base e , green is to base 10 , and purple is to base 1.7 . Each tick on the axes is one unit.

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Logarithms

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  1. Logarithms

  2. Logarithms • Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7. • 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.

  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 102 = 100). • Example log2 X = 8 28 = X X = 256

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

  5. Log 23.5 = 1.371 • Antilog 1.371 = 23.5 = 101.371

  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) 10log x = 100.25 X = 1.78

  8. Example 3 Review Log Rules • Logc (am) = m logc(a) • Solve for x 3x = 1000 • Log both sides to get rid of the exponent log 3x = 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. Solution Try It Out Problem 1 Solution

  12. Problem 2 • 3.5 = ln 5x

  13. Get rid of the ln by anti ln (ex) • e3.5 = eln 5x • e3.5 = 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 • e2.303= 10 ln 10 = 2.303 • e ln x = x

  16. SUMMARY

  17. In summary

  18. Simplify the following expression log59 + log23 + log26 • 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 106 = 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.)

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