Logarithms

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

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.

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.

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.

Review Log rules Log c (a m ) = m log c (a) Example log 2 X = = X X = log x = X “10 to the” is also the anti-log (opposite)

Example 2 Review Log rules Example 2 log X = 0.25 Raise both side to the power of log x = X = 1.78

Example 3 Review Log Rules 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

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.

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

Try It Out Problem 1 Solution Try It Out Problem 1 Solution log (x) 2 – log = 0

Simplify the following expression log log 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

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 (e x ) e 3.5 = e ln 5x e 3.5 = 5x 33.1 = 5x 6.62 = x

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

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 = log x 1/y = (1/y )log x ln = ln x 1/y =(1/y)ln x

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 = log x Why 2.303?

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 = or and log 10 = 1. So, substituting in we get / 1 = Voila!

In summary NumberExponential ExpressionLogarithm /10 = /100 = /1000 =

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.

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

More log sig fig examples log 2.7 x = The number has 2 significant figures, but its log ends up with 3 significant figures. ln 3.95 x 10 6 = = lots mantissa of 3

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 M? pH = -log [H+] = -log (5.0 x ) = - (-3.30) pH = 3.30

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+] = log [H+] = [H+] = inv log (-13.22) [H+] = 6.0 x M (2 sig. fig.)

QED Question?