Definition of Logarithm
Consider $a$ and $b$ two real positive numbers, with $a>0$, $b>0$, and $b\neq1$. If,
$\log_{b}{a}=x$ (logarithm of $a$ to base $b$), then $b^x=a$.
Properties of Logarithms
P.1: $b^{\log_{b}{a}}=a$
By the definition of logarithm, if $\log_{b}{a}=x$, then $b^x=a$.
Since $\log_{b}{a}=\log_{b}{a}$, then $b^{\log_{b}{a}}=a$
P.2: $\log_{b}{(ac)}=\log_{b}{a}+\log_{b}{c}$
Consider $\log_{b}{(ac)}=x$, and $\log_{b}{a}=y$, and $\log_{b}{c}=z$
Thus $b^x=ac$, and $b^y=a$, and $b^z=c$
$b^x=ac=b^yb^z=b^{(y+z)}$
$x=y+z$
P.3: $\log_{b}{(a^c)}=c\log_{b}{a}$
Consider $\log_{b}{(a^c)}=x$, and $\log_{b}{a}=y$
Then $b^x=a^c$, and $b^y=a$
$b^x={(b^y)}^c$
$b^x=b^{(cy)}$
$x=cy$
$\log_{b}{(a^c)}=c\log_{b}{a}$
P.4: $\log_{b}{(\frac{a}{c})}=\log_{b}{a}-\log_{b}{c}$
By P.2: $\log_{b}{(\frac{a}{c})}=\log_{b}{(ac^{-1})}=\log_{b}{a}+\log_{b}{c^{-1}}$
By P.3: $\log_{b}{a}+\log_{b}{c^{-1}}=\log_{b}{a}-\log_{b}{c}$
P.5: $\log_{b}{a}=\frac{\log_{c}{a}}{\log_{c}{b}}$
Consider: $\log_{b}{a}=x$, and $\log_{c}{a}=y$, and $\log_{c}{b}=z$
Then: $b^x=a$, and $c^y=a$, and $c^z=b$
$b^x=c^y$
$(c^z)^x=c^y$
$c^{(zx)}=c^y$
$zx=y$
$x=\frac{y}{z}$
P.6: $\log_{b}{a}=\frac{1}{\log_{a}{b}}$
By P.5: $\log_{b}{a}=\frac{\log_{a}{a}}{\log_{a}{b}}=\frac{1}{\log_{a}{b}}$
P.7: $\log_{(b^c)}{a}=\frac{1}{c}{\log_{b}{a}}$
By P.6: $\log_{(b^c)}{a}=\frac{1}{\log_{a}{(b^c)}}$
By P.3: $\frac{1}{\log_{a}{(b^c)}}=\frac{1}{c\log_{a}{b}}$
By P.6: $\frac{1}{c\log_{a}{b}}=\frac{1}{c}\log_{b}{a}$
Logarithmic Function
A logarithmic function is any function that can be rearranged in the form: $f(x)=\log_{b}{x}$, with $b>0$ and $b\neq1$.
When $b>1$, the function is monotonically increasing (red lines in the following graph);
When $0<b<1$, the function is monotonically decreasing (blue lines in the following graph);
The closer to 1 the base is, the steeper is the slope of the graph:
Graph of the Function $f(x)=\log_{b}{(ax+c)}$
The function $ax+c$ can be monotonically increasing (if $a>0$), or monotonically decreasing (if $a<0$).
The behavior of the logarithmic function $f(x)=\log_{b}{(ax+c)}$, depends on the behavior of the function $ax+c$ and on the value of the base $b$.
For $ax+c$ monotonically increasing ($a>0$):
if $b>1$, the logarithmic function is monotonically increasing;
if $0<b<1$, the logarithmic function is monotonically decreasing.
The graph below depicts the logarithmic function $f(x)=\log_{b}{(x+1)}$, thus $a=1$. In the blue line the base is equal to 2 ($b=2$); in the red line the base is 0.5 ($b=0.5$):
For $ax+c$ monotonically decreasing ($a<0$):
if $b>1$, the logarithmic function is monotonically decreasing;
if $0<b<1$, the logarithmic function is monotonically increasing.
The graph below depicts the logarithmic function $f(x)=\log_{b}{(-x+1)}$, thus $a=-1$. In the blue line the base is equal to 2 ($b=2$); in the red line the base is 0,5 ($b=0,5$):
Coefficient $c$ in the function $f(x)=\log_{b}{(ax+c)}$
Coefficient $c$ shifts the graph to the left or to the right (regardless of the value of the base):
if $c$ is positive, the graph shifts to the left;
if $c$ is negative, the graph shifts to the right.
The graph below depicts the functions:
$f(x)=\log_{2}{x}$,
$f(x)=\log_{2}{(x+1)}$,
$f(x)=\log_{2}{(x-1)}$.
Coefficient $a$ in the function $f(x)=\log_{b}{(ax+c)}$
Coefficient $a$ shifts the graph up and down, depending on the value of the base $b$.
For $b>1$:
if $a>1$, the graph shifts up;
if $0<a<1$, the graph shifts down.
The graph below depicts the functions:
$f(x)=\log_{2}{x}$,
$f(x)=\log_{2}{(2x)}$,
$f(x)=\log_{2}{(0,5x)}$.
For $0<b<1$:
if $a>1$, the graph shifts down;
if $0<a<1$, the graph shifts up.
The graph below depicts the functions:
$f(x)=\log_{0,5}{x}$,
$f(x)=\log_{0,5}{(2x)}$,
$f(x)=\log_{0,5}{(0,5x)}$.
When $a<0$, the graph is reflected across the y-axis.
The graph below depicts the functions:
$f(x)=\log_{2}{x}$,
$f(x)=\log_{2}{(-x)}$.
Graph of the Function $f(x)=\log_{b}{(ax^2+cx+e)}$
The graph of the function $ax^2+cx+e$ is a parabola. Thus part of it is increasing, part is decreasing. In order to understand the behavior of the logarithmic function $f(x)=\log_{b}{(ax^2+cx+e)}$, we have to consider:
If $b>1$,
when $ax^2+cx+e$ is increasing, the logarithmic function is increasing;
when $ax^2+cx+e$ is decreasing, the logarithmic function is decreasing.
If $0<b<1$,
when $ax^2+cx+e$ is increasing, the logarithmic function is decreasing;
when $ax^2+cx+e$ is decreasing, the logarithmic function is increasing.
Consider, for example, the function $f(x)=\log_{2}{(x^2+x-2)}$.
Between -2 and 1, $x^2+x-2$ is negative; thus $f(x)=\log_{2}{(x^2+x-2)}$ has no solution.
For $x<-2$, $x^2+x-2$ is decreasing; thus $f(x)=\log_{2}{(x^2+x-2)}$ is also decreasing.
For $x>1$, $x^2+x-2$ is increasing; thus $f(x)=\log_{2}{(x^2+x-2)}$ is also increasing.
The graph below depicts $f(x)=\log_{2}{(x^2+x-2)}$:
The graph below depicts $f(x)=\log_{2}{(x^2+x-2)}$:
Mantissa
For any real number:
$10^c<=x<10^{(c+1)}$
$\log_{10}{(10^c)}<=\log_{10}{x}<\log_{10}{10^{(c+1})}$.
$c<=\log_{10}{x}<c+1$
$c<=\log_{10}{x}<c+1$
Therefore, the logarithm of any real number can be writen as $c+m$, where:
$c$ is an integer, called Characteristic, and
$m$ is an irrational number between 0 and 1 ($0<=m<1$) called Mantissa.
Find Practice Tests in the following links: (not available yet)
SAT Practice Tests - Logarithmic Functions and Equations
Additional Practice Tests - Logarithmic Functions and Equations
$c$ is an integer, called Characteristic, and
$m$ is an irrational number between 0 and 1 ($0<=m<1$) called Mantissa.
Solved SAT Practice Tests
Find Practice Tests in the following links: (not available yet)
SAT Practice Tests - Logarithmic Functions and Equations
Additional Practice Tests - Logarithmic Functions and Equations
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