# Logarithms

The function f(x) = 2x is called an exponential function because the variable, x, is the exponent. In general, exponential functions are of the form f(x) = ax, where a is a positive constant. The inverse of an exponential function is called a logarithmic function. Therefore, the inverse of f(x) = ax is the logarithmic function with base a, such that y = logax ↔ ay = 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 [10x] 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: 