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

f(x)=\log _{b}x       and      g(x)=\ln 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
-1undefined
0undefined
10
20.3
30.477

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

The x-intercept is at  (1,0)

Domain: { x \epsilon R | x>0 }

Range: { y\epsilon R }

log graph

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

x

y
-1undefined
0undefined
10
20.693
31.0986

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

The x-intercept is at  (1,0)

Domain: { y\epsilon R }

natural log graph

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)=2logxf(x)=\frac{1}{2}logx
f(x)=log2xf(x)=log\frac{1}{2}x
f(x)=-logxf(x)=log(-x)
f(x)=log(x-2)f(x)=log(x+2)
f(x)=logx-2f(x)=logx+2

State transformations in order and graph

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)

transformed log function graph

Review Exponential Functions

Applications of Logs

Log laws (base 10 logarithm)

loga + log b = log(ab)

loga - logb=log (\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 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

pH Scale - Carnivorous Plant Resource

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

Solving Logarithmic Equations Quiz