1. Logarithmic Functions
Digital Lesson
Logarithmic Functions

2. Definition: Logarithmic Function
For x  0 and 0  a  1,
y = loga x if and only if x = a y.
The function given by f (x) = loga x is called the logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form:
y = loga x is equivalent to x = a y
A logarithm is an exponent!
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
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Definition: Logarithmic Function

3. Examples: Write Equivalent Equations
Examples: Write the equivalent exponential equation
and solve for y.
Solution
Equivalent Exponential Equation
Logarithmic Equation
y = log216
16 = 2y
16 = 24  y = 4
y = log2( )
= 2 y
= 2-1 y = –1
y = log416
16 = 4y
16 = 42  y = 2
y = log51
1 = 5 y
1 = 50  y = 0
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Examples: Write Equivalent Equations

4. Common Logarithmic Function
The base 10 logarithm function f (x) = log10 x is called the common logarithm function.
The LOG key on a calculator is used to obtain common logarithms.
Examples: Calculate the values using a calculator.
Function Value
Keystrokes
Display
log10 100
LOG 100 ENTER 2
log10( )
LOG ( ) ENTER –
log10 5
LOG ENTER
log10 –4
LOG –4 ENTER
ERROR
no power of 10 gives a negative number
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Common Logarithmic Function

5. Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. loga ax = x and alogax = x inverse property
4. If loga x = loga y, then x = y. one-to-one property
Examples: Solve for x: log6 6 = x
log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log79
7log79 = 9 property 3
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Properties of Logarithms

6. horizontal asymptote y = 0
Graph f (x) = log2 x
Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. x y
y = 2x
y = x 8 3 4 2 1 –1 –2 2x x
horizontal asymptote y = 0
y = log2 x
x-intercept
(1, 0)
vertical asymptote
x = 0
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Graph f(x) = log2 x

7. Example: Graph the common logarithm function f(x) = log10 x.
0.602
0.301 –1 –2
f(x) = log10 x 10 4 2 x y x 5 –5
by calculator
f(x) = log10 x
(0, 1) x-intercept
x = 0 vertical asymptote
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Example: f(x) = log0 x

8. Graphs of Logarithmic Functions
The graphs of logarithmic functions are similar for different values of a f(x) = loga x (a  1)
y-axis
vertical
asymptote x y
Graph of f (x) = loga x (a  1)
y = a x
y = x
range
1. domain
y = log2 x
2. range
3. x-intercept (1, 0)
4. vertical asymptote
domain
x-intercept
(1, 0)
5. increasing
6. continuous
7. one-to-one
8. reflection of y = a x in y = x
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Graphs of Logarithmic Functions

9. Natural Logarithmic Functiony x 5 –5
y = ln x
The function defined by f(x) = loge x = ln x
(x  0, e )
is called the natural logarithm function.
y = ln x is equivalent to e y = x
Use a calculator to evaluate: ln 3, ln –2, ln 100
Function Value
Keystrokes
Display
ln 3
LN 3 ENTER
ln –2
LN –2 ENTER
ERROR
ln 100
LN 100 ENTER
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Natural Logarithmic Function

10. Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x and eln x = x inverse property
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
inverse property
inverse property
property 2
property 1
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Properties of Natural Logarithms

11. Example: Carbon Dating
Example: The formula (t in years) is used to estimate the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is How old is it?
original equation
multiply both sides by 1012
take the natural log of both sides
inverse property
To the nearest thousand years the charcoal is 57,000 years old.
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Example: Carbon Dating