A Logarithmic Function is an inverse function of an exponential function.

$f(x)=\log&space;_{b}x$       and      $g(x)=\ln&space;x$

Logarithmic functions are widely used to represent quantities that spread out over a wide range (earthquake magnitude, sound level, etc).

Taking the logarithm of a number produces the exponent to which the base of the logarithm was originally raised.

Let’s go over how to graph logarithmic functions and how to solve logarithmic equations.

## Graphing Logarithmic Functions

$f(x)=logx$   (base 10, from $g(x)=10^x$)

 $x$ $y$ -1 undefined 0 undefined 1 0 2 0.3 3 0.477

There are no $y$-intercept, as there is a Vertical Asymptote (VA) at $x=0$

The $x$-intercept is at  (1,0)

Domain: { $x&space;\epsilon&space;R&space;|&space;x>0$ }

Range: { $y\epsilon&space;R$ }

$f(x)=lnx$  (base $e$, natural logarithm from $g(x)=e^x$)

 $x$ $y$ -1 undefined 0 undefined 1 0 2 0.693 3 1.0986

There are no $y$-intercept, as there is a Vertical Asymptote (VA) at $x=0$

The $x$-intercept is at  (1,0)

Domain: { x&space;\epsilon&space;R|&space;x>0″ alt=”x \epsilon R| x>0″ align=”absmiddle”> }; Range: { }

### Transformations of Log Functions

$f(x)=alog(k(x-c))+d$

$a$ – vertical stretch or compression, reflection in the $x$-axis

$k$ – horizontal stretch or compression, reflection in the $y$-axis

$c$ – horizontal translation left or right

$d$ – vertical translation up or down

 $f(x)=2logx$ $f(x)=\frac{1}{2}logx$
 $f(x)=log2x$ $f(x)=log\frac{1}{2}x$
 $f(x)=-logx$ $f(x)=log(-x)$
 $f(x)=log(x-2)$ $f(x)=log(x+2)$
 $f(x)=logx-2$ $f(x)=logx+2$

#### $f(x)=-2log(x-1)+3$

• vertical compression by a factor of 2
• reflection in the $x$ – axis

• horizontal translation 1 unit left

• vertical translation 3 units up

Vertical Asymptote (VA): $x=1$

$y$ – intercept: $f(0)=-2log(0-1)+3$      –    NONE

$x$ – intercept(s):

In order to determine the $x$ – intercepts, set the equation to = 0, then solve it for $x$ by performing inverse operations in the reverse BEDMAS order.

At the end, the equation would need to be changed to an exponential form in order to be solved.

$0=-2log(x-1)+3$

$-3=-2log(x-1)$

$\frac{-3}{-2}=log(x-1)$

$\frac{3}{2}=log(x-1)$

at this point, in order to solve this equation, represent it in its exponential form, where base is 10 and the exponent is $\frac{3}{2}$ :

$10^{\frac{3}{2}}=x-1$

$x=10^\frac{3}{2}+1$

$x=32.6$

Therefore, the $x$ – intercept is at (32.6, 0)

Review Exponential Functions

## Applications of Logs

#### Log laws (base 10 logarithm)

$loga&space;+&space;log&space;b&space;=&space;log(ab)$

$loga&space;-&space;logb=log&space;(\frac{a}{b})$

$loga^b=bloga$

$log1=0$

Laws of Logarithms Quiz

Example 1: A new car is purchased for \$20,000. The car’s value after $t$ years is given by  $V=20,000(0.8)^t$. How long will it take for the car to be worth half of its purchase value?

Solution:

$10,000=20,000(0.8)^t$

$\frac{10,000}{20,000}=0.8^t$

$0.5=0.8^t$     convert into logarithmic form

$log0.5=tlog0.8$

$t=\frac{log0.5}{log0.8}$

$t=3.1$

Therefore, it would take just over 3 years for the car do depreciate to half of its original value.

Example 2: On the Richter scale, the magnitude of the earthquake in City A is 2.7 and the magnitude of the earthquake in City is  . How many times stronger is the earthquake in City A?

Formula to compare two earthquakes: $M_{2}-M_{1}=log(\frac{I_{2}}{I_{1}})$ where M is the magnitude and I is the intensity.

Solution:

$2.7-1.3=log(\frac{I_{A}}{I_{B}})$

$1.4=log(\frac{I_{a}}{I_{b}})$    convert into exponential form with base 10

$\frac{I_{A}}{I_{B}}=10^{1.4}$

$\frac{I_{A}}{I_{B}}\approx&space;25$

Therefore, the earthquake in City A is approximately 25 times stronger than the earthquake in City B.

Example 3: Tomato juice has a hydronium ion concentration of approximately 0.0001 mol/L. What is its pH? (acidity level)

Formula to determine the pH: $pH=-log[H^{+}]$

Solution:

$pH=-log(0.0001)$

$pH=4$

Since the pH level of tomato juice is 4, it is acidic.

Solving Logarithmic Equations Quiz