Writing An Expression As A Single Logarithm Calculator

Math Skills Review
Logarithms

Two kinds of logarithms are often used in chemistry: common (or Briggian) logarithms and natural (or Napierian) logarithms. The power to which a base of 10 must be raised to obtain a number is called the common logarithm (log) of the number. The power to which the base e (e = 2.718281828.......) must be raised to obtain a number is called the natural logarithm (ln) of the number.

In simpler terms, my 8th grade math teacher always told me: LOGS ARE EXPONENTS!! What did she mean by that?

Using log₁₀ ("log to the base 10"):
log₁₀100 = 2 is equivalent to 10² = 100
where 10 is the base, 2 is the logarithm (i.e., the exponent or power) and 100 is the number.
Using natural logs (log_e or ln):
Carrying all numbers to 5 significant figures,
ln 30 = 3.4012 is equivalent to e^3.4012 = 30 or 2.7183^3.4012 = 30
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? 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, the number has to be 2.303. Voila!
Historical note: Before calculators, we used slide rules (a tool based on logarithms) to do calculations requiring 3 significant figures. If we needed more than 3 signficant figures, we pulled out our lengthy logarithm tables. Anyway, enough history......

The rest of this mini-presentation will concentrate on logarithms to the base 10 (or logs). One use of logs in chemistry involves pH, where pH = -log₁₀ of the hydrogen ion concentration.

FINDING LOGARITHMS

Here are some simple examples of logs.

Number	Exponential Expression	Logarithm
1000	10³	3
100	10²	2
10	10¹	1
1	10⁰	0
1/10 = 0.1	10^-1	-1
1/100 = 0.01	10^-2	-2
1/1000 = 0.001	10^-3	-3

To find the logarithm of a number other than a power of 10, you need to use your scientific calculator or pull out a logarithm table (if they still exist). On most calculators, you obtain the log (or ln) of a number by

entering the number, then
pressing the log (or ln) button.

Example 1: log 5.43 x 10¹⁰ = 10.73479983...... (way too many significant figures)
Example 2: log 2.7 x 10^-8 = -7.568636236...... (too many sig. figs.)

So, let's look at the logarithm more closely and figure out how to determine the correct number of significant figures it should have.

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. So in the above examples:

Example 1: log 5.43 x 10¹⁰ = 10.735
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.
Example 2: log 2.7 x 10^-8 = -7.57
The number has 2 significant figures, but its log ends up with 3 significant figures.

Natural logarithms work in the same way:

Example 3: ln 3.95 x 10⁶ = 15.18922614... = 15.189

Application to pH problems:

pH = -log (hydrogen ion concentration) = -log [H⁺]

Example 4: 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) = 3.30

FINDING ANTILOGARITHMS (also called Inverse Logarithm)

Sometimes we know the logarithm (or ln) of a number and must work backwards to find the number itself. This is called finding the antilogarithm or inverse logarithm of the number. To do this using most simple scientific calculators,

enter the number,
press the inverse (inv) or shift button, then
press the log (or ln) button. It might also be labeled the 10^x (or e^x) button.

Example 5: log x = 4.203; so, x = inverse log of 4.203 = 15958.79147..... (too many significant figures)
There are three significant figures in the mantissa of the log, so the number has 3 significant figures. The answer to the correct number of significant figures is 1.60 x 10⁴.
Example 6: log x = -15.3;
so, x = inv log (-15.3) = 5.011872336... x 10^-16 = 5 x 10^-16 (1 significant figure)

Natural logarithms work in the same way:

Example 7: ln x = 2.56; so, x = inv ln (2.56) = 12.93581732... = 13 (2 sig. fig.)

Application to pH problems:

pH = -log (hydrogen ion concentration) = -log [H⁺]

Example 8: 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.)

CALCULATIONS INVOLVING LOGARITHMS

Because logarithms are exponents, mathematical operations involving them follow the same rules as those for exponents.

Common Logarithm	Natural Logarithm
log xy = log x + log y	ln xy = ln x + ln y
log x/y = log x - log y	ln x/y = ln x - ln y
log x^y = y log x	ln x^y = y ln x
log = log x^1/y = (1/y )log x	ln = ln x^1/y =(1/y)ln x

Example 9: log 5.0 x 10⁶ = log 5.0 + log 10⁶ = 0.70 + 6 = 6.70
Hint: This is an easy way to estimate the log of a number in scientific notation!
Example 10: log (154/25) = log 154 - log 25 = 2.188 - 1.40 = 0.788 = 0.79 (2 sig. fig.)
Example 11: log (5.46 x 10^-3)⁶ = 6 log 5.46 x 10^-3 = 6 x (-2.263) = -13.58
Example 12: log (4.35)^1/4 = 1/4 log 4.35 = 0.160

QUIZ: Give the correct answer to the correct number of significant figures.

Question 1	log 23.0 = ?
Question 2	What is the pH of an aqueous solution in which [H⁺] = 2.7 x 10^-3 M?
Question 3	Find x if log x = 10.23.
Question 4	If the pH of an aqueous solution is 6.52, what is the concentration of the hydrogen ion?
Question 5	log (1.2 x 10⁶)³ = ?