understandin the term logarithm and log. 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 
$$\log_{2x+5}(10x^2+29x+10)=5−\log_{5x+2}(4x^2+20x+25)$$
 A: 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!
A: 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
