Functions with Base b

Key Questions

What are some identity rules for logarithms?

Log/Exp Inverse Properties

$b^{{log}_{b} x} = x$ $b^{log_b x}=x$

${log}_{b} b^{x} = x$ $log_b b^x=x$

Other Log Properties

${log}_{b} (x \cdot y) = {log}_{b} x + {log}_{b} y$ $log_b(xcdot y)=log_b x+log_b y$

${log}_{b} (\frac{x}{y}) = {log}_{b} x - {log}_{b} y$ $log_b(x/y)=log_b x-log_b y$

${log}_{b} x^{r} = r {log}_{b} x$ $log_b x^r=r log_b x$

I hope that this was helpful.

Wataru · · Oct 20 2014
What is the exponent rule of logarithms?

Answer:

${log}_{a} (m^{n}) = n {log}_{a} (m)$ $log_(a)(m^(n)) = n log_(a)(m)$

Explanation:

Consider the logarithmic number ${log}_{a} (m) = x$ $log_(a)(m) = x$ :

${log}_{a} (m) = x$ $log_(a)(m) = x$

Using the laws of logarithms:

$\Rightarrow m = a^{x}$ $=> m = a^(x)$

Let's raise both sides of the equation to $n$ $n$ th power:

$\Rightarrow m^{n} = {(a^{x})}^{n}$ $=> m^(n) = (a^(x))^(n)$

Using the laws of exponents:

$\Rightarrow m^{n} = a^{x n}$ $=> m^(n) = a^(xn)$

Let's separate $x n$ $xn$ from $a$ $a$ :

$\Rightarrow {log}_{a} (m^{n}) = x n$ $=> log_(a)(m^(n)) = xn$

Now, we know that ${log}_{a} (m) = x$ $log_(a)(m) = x$ .

Let's substitute this in for $x$ $x$ :

$\Rightarrow {log}_{a} (m^{n}) = n {log}_{a} (m)$ $=> log_(a)(m^(n)) = n log_(a)(m)$

Tazwar Sikder · 1 · Sep 16 2016
What does a logarithmic function look like?

Answer:

The reflection of the exponential function on the axis $y = x$ $y=x$

Explanation:

Logarithms are the inverse of an exponential function, so for $y = a^{x}$ $y=a^x$ , the log function would be $y = {log}_{a} x$ $y=log_ax$ .

So, the log function tell you what power $a$ $a$ has to be raised to, to get $x$ $x$ .

Graph of $ln x$ $lnx$ :
graph{ln(x) [-10, 10, -5, 5]}

Graph of $e^{x}$ $e^x$ :
graph{e^x [-10, 10, -5, 5]}

Karthik N · 1 · May 1 2018