# Help with math questions with logarithms

## Logarithm / Logarithm (Calculate)

**Many students have little knowledge of the logarithm and do not understand the relationships between the different forms: What was the natural logarithm and what is the decadic logarithm? The following article is intended to answer these questions using examples.**

Many of you have surely already had to solve equations or even entire systems of equations. For example, one had to solve an equation of the form 2 + 5x = 0 for x. This was solved through addition, subtraction, multiplication and division. But suppose you should y = 2^{x} solve for x. So what? The solution is: use logarithm. That is exactly what we take care of in this section. Before doing this, however, you should make sure that you are familiar with the following topics. If you still have problems with these, follow the links. Everyone else can get started with the logarithm right away.

**Logarithm for base 2: logarithm of two**

Let's look again at the example from above: y = 2^{x}. This equation should now be solved for x. For this reason we log the equation. It looks like this:

y = 2^{x} | | logarithmize |

log_{2}y = x |

As with any equation, what you do on the left must also be done on the right. Thus the logarithm is applied on both sides. log_{2}y = x means: The logarithm of y to base 2 is equal to x. So you have to think about the following: What exponent x do I need, with which the number 2 has to be raised to the power to get y? The example above showed the logarithm of two, because the basis was a 2. Let's look at a few more examples for a better understanding.

**Examples:**

- log
_{2}16 = 4, because 2^{4}= 16 - log
_{2}1024 = 10, because 2^{10}= 1024

**Further formulas / calculation rules**

There are a number of other rules for the logarithm, which we would like to introduce to you here with an example:

Table scrollable to the rightCalculation rule | example |

log_{a} (u * v) = log_{a}u + log_{a}v | log_{2} (4 x 8) = log_{2}4 + log_{2}8 = 2 + 3 = 5 |

log_{a} (u: v) = log_{a}u - log_{a}v | log_{3} (81: 9) = log_{3}81 - log_{3}9 = 4 - 2 = 2 |

log_{a}u^{n} = n log_{a}u | log_{5}125^{4} = 4 x log_{5}125 = 4 · 3 = 12 |

**Natural and decadic logarithms **

Before we start with the natural logarithm, a little reminder: An E-function has the form y = e^{ax}, for example y = e^{2x} or y = e^{5x}. The e is the so-called Euler's number, which occurs in many scientific and technical functions. In the equation, e = 2.718 ... We now insert the rounded value 2.718 for the e into the equation. Alternatively, many pocket calculators have an "e" directly on them so that they can calculate.

Earlier in the article, we already worked with base 2 as well as with the general base in the formulas (see the calculation rules and examples in the table). In mathematics, two different names were given for base 10 and base e.

**Natural logarithm:**

If one has the base e, this leads to the natural logarithm. This looks like this, for example: y = log_{e}x. There is also an abbreviated form y = lnx for this. Which spelling you prefer is up to you (or is given by the math teacher).

**Note: log _{e}x = lnx**

**Decadal logarithm:**

If, on the other hand, you have the base 10, this leads to the decadic logarithm or also called the logarithm of ten. The form: y = log_{10}r. There is also an abbreviation here: lg r.

**Note: log _{10}r = lgr**

**More about the logarithm:**

**More links: **

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