understandin the term logarithm and log.

Question

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

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

Answer 1

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!

Answer 2

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