Logarithms

The function f(x) = 2^x is called an exponential function because the variable, x, is the exponent. In general, exponential functions are of the form f(x) = a^x, where a is a positive constant. The inverse of an exponential function is called a logarithmic function. Therefore, the inverse of f(x) = a^x is the logarithmic function with base a, such that y = log_ax ↔ a^y = x. In many science applications, we are interested in base 10 logarithms.

Evaluating Logarithms

In science it is important to be able to evaluate log functions, as they come up in many applications. To evaluate the log (base 10) of a number, type the number into your calculator then hit the [log] key. Some calculators are slightly different; therefore, it is important for students to be familiar with their own calculator. It is also important in many situations to be able to determine the antilog of a log value. To evaluate the antilog of a number, type the number into the calculator and hit the [10^x] key. 

Example: Find x in each of the following equations: x = log₁₀3.6 and log₁₀x ₌ 6.75.

Solution:

x = log₁₀3.6                                         log₁₀x ₌ 6.75.

x = 0.556                                                      x= 5.62x10⁶

Example: Convert [H+] = 3.5x10^-3 M to pH.

Solution:

pH = -log[H⁺]

pH = -log [3.5 x 10^-3]

pH = 2.4

Example: What value of [H+] corresponds to a pH of 4.3?

Solution:

pH = -log [H+]

4.3 = -log [H+]

-4.3  = log [H+]

[H+] = 10^-4.3

[H+] = 5.0 x 10^-5 M

Example: