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How should I make someone understand the term log or logarithm and how its values are determined who is standard VI? like:

$$\log_2 8 = 3$$

or

$$\log_{2x+5}(10x^2+29x+10)=5−\log_{5x+2}(4x^2+20x+25)$$

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  • $\begingroup$ I should probably start with the number of zero's in $10^n$, $n$ being a positive integer. This at least introduces the logarithmic scale. $\endgroup$ – Claude Leibovici Apr 18 '17 at 9:57
  • $\begingroup$ What does "who is standard VI?" mean? Mathematics is not as standardised as some subjects (e.g. chemistry) so there are some questions that do not have definitive answers. Some terms are almost universally agreed and others vary a lot. With logs, the base varies: 10, e, and 2 are all quite common. $\endgroup$ – badjohn Apr 18 '17 at 11:42
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This explanation will contain both a small rigorous and elementary explanation of your following query.

$\mathbf{Rigerous \ Explanation:}$

The logarithmic function, denoted as $log(y)$ is the geometric area under a $\frac 1t$ vs. $t$ curve, mathematically equivalent to $log(y)=\int_1^y\frac 1tdt$, bounded for $t\in[1,y]$ and the line $y=0$. Furthermore, by taking the inverse of $log(y)$, we generate an analogous (exponential) function notated as $exp(x)$ or $e^x$, geometrically equivalent to the reflection of $log(y)$ about the line $y=x$.

In fact, in real analysis, $log(y)$ can be defined by considering an arbitrary continuous function which satisfies $f(x+y)=f(x)f(y)$ and $f(0)=1$, for $x,y>0$, and then applying it to the definition of a derivative. This is exactly how the result $log(y)=\int_1^y\frac 1tdt$ is derived.

Furthermore, the generation of such a function proves useful when trying to determine an expression, such that its value is its own derivative, called Euler's constant, or $e$. As a note, mathematicians often find it convenient to define the logarithm, as the inverse of the exponential function, as stated above.

$\mathbf{Elementary \ Explanation:}$

Consider the following more general exponential function:

$f(x)=b^x$, where $b$ is called the "base" and $x$ the "exponent" or "index".

Furthermore, when $b=e\approx2.718$, then the inverse of $y=e^x$ is $log(y)$, or equivalently $ln(y)$ (called the natural logarithm). As a note, if not specified, a logarithm without a base, or notated as $log(y)$, is assumed to have base $e$.

We can extend this notion further, however, by considering other bases other than $e$, such as binary ($2$) or $10$. Our value of $e$ proves very useless in this regard, as it bridges base conversion between exponential functions, that is:

$$b^x=e^{xlogb}$$

This comes by definition, but the equivalence is seen by simply taking the logarithm of both sides.

Let's consider the equality $1000=10^3$

Then the corresponding logarithmic equivalent would be:

$$log_{10}1000=3$$

The above expression denotes how many times the base $10$, will be multiplied by itself, to attain a value of $1000$. As you can see, this is quite the opposite to $1000=10^3$, as it reverses its initial exponential operation.

Furthermore, there are several algebraic properties of logarithms, as shown below:

For $x,y,r>0$

$$log_b(xy)=log_b(x)+log_b(y)$$

$$log_b(\frac xy)=log_b(x)-log_b(y)$$

$$log_b(x^r)=rlog_b(x)$$

$$log_b(x)=\frac {log(x)}{log(b)}$$

The proofs for these are pretty straightforward, and can be looked up online.

Please note, this is only a small introduction to logarithms/exponential functions, and a lot more can be said!

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logorithm and exponential function are inverse of each other $$a^{x} = y$$ then it implies that $$\log_{a}^{y} = x$$ "a" is called base of the logorithm ... I think the property used in your problem is change of base property of log

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