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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 • Logarithms, or "logs", are a simple way of expressing numbers in terms of a single base. • Common logs are done with base ten, but some logs ("natural" logs) are done with the constant "e" as their base. • The log of any number is the power to which the base must be raised to give that number.

  4. In other words, log(10) is 1 and log(100) is 2 (because 102 = 100). Logs can easily be found for either base on your calculator. Usually there are two different buttons, one saying "log", which is base ten, and one saying "ln", which is a natural log, base e. It is always assumed, unless otherwise stated, that "log" means log10.

  5. Chem? • Logs are commonly used in chemistry. • The most prominent example is the pH scale. • The pH of a solution is the -log([H+]), where square brackets mean concentration.

  6. Review Log rules • Logc (am) = m logc(a) • Example log2 X = 8 28 = X X = 256 • 10log x = X • “10 to the” is also the anti-log (opposite)

  7. Example 2 Review Log rules • Example 2 log X = 0.25 • Raise both side to the power of 10 10log x = 100.25 X = 1.78

  8. Example 3 Review Log Rules • 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 • The log of one number times the log of another number is equal to the log of the first plus the second number. • Similarly, the log of one number divided by the log of another number is equal to the log of the first number minus the second. • This holds true as long as the logs have the same base.

  10. Multiplying and Dividing logs • Log (a * b) = log a + log b • Log (a / b) = log a – log b

  11. Try It Out Problem 1 Solution log (x)2 – log 10 - 3 = 0 Try It Out Problem 1 Solution

  12. 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

  13. Solve the following problem. 7 = ln5x + ln(7x-2x) • Simplify! 7 = ln 5x + ln 5x (PEMDAS) • Log (ln) rules 7 = 2 ln 5x Adding goes to mult. when you remove an ln. • (7 / 2) = ln 5x • 3.5 = ln 5x • 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 • Negative powers of 10 may be fitted into the system of 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. SUMMARY

  16. 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?

  17. 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!

  18. In summary

  19. Sig Figs and logs • For any log, the number to the left of the decimal point is called the characteristic, and the number to the right of the decimal point is called the mantissa. • The characteristic only locates the decimal point of the number, so it is usually not included when determining the number of significant figures. • The mantissa has as many significant figures as the number whose log was found.

  20. SHOW ME! • log 5.43 x 1010 = 10.735 • The number has 3significant figures, but its log ends up with 5 significant figures, since the mantissa has 3 and the characteristic has 2. • ALWAYS ASK THE MANTISSA!

  21. 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 = 15.18922614... = 15.189 3 lots mantissa of 3

  22. 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

  23. 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.)

  24. QED • Question?

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