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CALCULATE

CALCULATE

** Logarithm (LOG) calculator** is an online math calculator that calculates the log value for the positive real number with respect to the given or natural base values (positive, not equal to $1$). Using this calculator, we will understand methods of how to find the logarithm of any number with respect to the given base.

It is necessary to follow the next steps:

- Enter the number and the base of logarithm. These values must be positive real numbers or parameter. The base of logarithm can not be $1$;
- Press the "
**CALCULATE**" button to make the computation; - Logarithm calculator will give the logarithm of the positive real number number with the positive base not equal to $1$.

To recall, exponential functions are very interesting as models for description of natural processes, physical magnitudes, as well as economic and social problems. A function $f(x)=a^x$, where $a>0$ is called an exponential function. The positive constant $a$ is called the base of the exponential function.
The most commonly encountered base of exponential function is the number $e=2.7182818...$ This number is called **Euler's number**. The exponential function with the base $e$ is $f(x)=e^x$ and it is often called the natural exponential function. The exponential function $f(x)=a^x$, for $a>0, a\ne 1$, is bijection so it has an inverse function. The inverse function $g(x)$ of $f(x)$ is
$$g(x)=\log_a x$$
where $a>0, a\ne 1$.
From the definition, it holds that $f(x)=\log_a x$ if and only if $x=a^{f(x)}$.
This inverse function of the exponential function is called \underline{the logarithmic function} for the base $a$.

Pierre-Simon Laplace called logarithms: "an admirable artifice which, by reducing to a few days the labor of many months, doubles the life of the
astronomer, and spares him the errors and disgust that are inseparable from long calculations".

Log_{b} x | |
---|---|

Log_{10} 2 | 0.301 |

Log_{2} 2 | 1 |

Log_{2} 10 | 3.3219 |

Log_{2} 5 | 2.3219 |

Log_{2} 20 | 4.3219 |

Log_{2} 100 | 6.6439 |

Log_{2} 6 | 2.585 |

Log_{2} 1000 | 9.9658 |

Log_{2} 15 | 3.9069 |

Log_{2} 32 | 5 |

Log_{2} 7 | 2.8074 |

Log_{10} 20 | 1.301 |

Log_{10} 10 | 1 |

Log_{10} 4 | 0.6021 |

Log_{10} 100 | 2 |

Log_{10} 5 | 0.699 |

Log_{10} 13 | 1.1139 |

Log_{10} 6 | 0.7782 |

Log_{10} 2 | 0.301 |

Log_{10} 1 | 0 |

Log_{10} 1000 | 3 |

ln(2) | 0.6931 |

ln(0) | -∞ |

ln(3) | 1.0986 |

ln(20) | 2.9957 |

ln(10) | 2.3026 |

ln(1) | 0 |

ln(e) | 1 |

There are two bases most commonly used, the natural logarithm and the common logarithm.The logarithm with the base of $e=2.7182818...$ is called natural logarithm. The first mention of the natural logarithm was by Nicholas Mercator, a mathematics teacher at the University of Copenhagen. The natural logarithm is denoted by $\ln x$. The function of the natural logarithm is cited in formulas for compound interest and economic growth rate.

The logarithm with the base of $10$, i.e. $\log_{10} x$, is denoted as ${\rm lg} \;x$. It is also known as the **decimal logarithm** or the standard logarithm.

The best way to calculate log of numbers was by using log tables. Invented in the early 1600s century by John Napier (Description of the Wonderful Rule of Logarithms 1614), log tables were a crucial tool used in computations prior to the advent of electronic calculators.

From the definitions, it is easy to conclude that logarithms are defined only for positive real numbers.
To find a logarithm with an arbitrary base using only natural and decimal logarithms, we need to apply the following rule
$$\log_a x=\frac{\ln x}{{\rm lg} \;x}$$
Every logarithmic equation corresponds to an equivalent exponential equation.
In other words, the logarithm of a number to a given base is the exponent by which base has to be raised to produce that number. For instance, $\log_3 81=4$ because $3^4=81$.

More precisely, if $x$ and $a$ are two positive real numbers and for some real number $b$, if
$a^b = x$ then $b$ is the logarithm of $x$ to the base $a$, and is
written as $$b=\log_a x $$
There are some interesting properties of logarithms that deserves to be mentioned here:

- Logarithms of the same number to different bases are different;
- Logarithm of $1$ to any base $a>0, a\ne 1$ is $0$;
- Logarithm of $1$ to any nonzero base is zero;
- Logarithm of any nonzero positive number to the same base is $1$;
- $\log_a b^n =n\log_ab$, $a\ne 1$;
- $\log_a(b\times c)=\log_ab+\log_ac$, $a\ne 1$;
- $\log_a\Big(\frac{b}{c}\Big)=\log_a b-\log_ac$, $a\ne 1$;
- $\log_ab=\frac{\log_cb}{\log_ca},\; c>0,a,c\ne1$;
- $\log_{a^n}b=\frac 1n\log_ab$, $a\ne 1$;
- $\log_{a}{b^n}=n\log_ab$, $a\ne 1$;

for positive real numbers $a,b$ and $c$.

Understanding the concepts of logarithms can help us to solve many mathematical problems. First of all, logarithms are related with arithmetic and geometric progression. Logarithms arise in probability theory: for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. This can be described by the law of the iterated logarithm. The logarithmic function is one of the most useful functions in mathematics, with applications to mathematical models throughout the physics, biologic and other fields of science and life. Logarithms also have practical uses in many fields of life and nature. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. So it can be represented by a logarithmic spiral.

**Practice Problem 1 : **

Atmospheric pressure $P$ decreases as the altitude $h$ above sea level increases. Atmospheric pressure
at altitude can be represented by the logarithmic function
$$\log_{2.5}\frac P{1.45}=-1.5h$$
The elevation of Paris is about $1$ kilometer. Find the atmospheric pressure in Paris.

**Practice Problem 2 : ** How long would it take for $100,000$ grams of chemical compound, with a half-life of $20$ days, to decay to $150$ grams?

The Logarithm Calculator, formula, example calculation, real world problems and practice problems would be very useful for grade school students (K-12 education) to understand the concept of exponents and logarithm. This concept can be of significance in calculus, algebra, probability and many other fields of science and life.