Transcript Evaluating Logarithms

Section 3.3a!!!
First, remind me…
What does the horizontal line test tell us???
More specifically, what does it tell us about the function
f  x   bx
This function has an inverse
that is also a function!!!
This inverse is called the
logarithmic function with base b.
Notation:
f
1
 x   logb x
Changing Between Logarithmic
and Exponential Form
If x > 0 and 0 < b = 1, then
y  logb x
if and only if
b x
y
Important Note: The “linking statement” says that
a logarithm is an exponent!!!
Basic Properties of Logarithms
For 0 < b = 1, x > 0, and any real number y,
1.
2.
3.
4.
logb 1  0 because b  1
1
logb b  1 because b  b
y
y
y
log b b  y because b  b
0
b
logb x
x
because
logb x  logb x
Evaluating Logarithms
Evaluate each of the following.
1.
2.
3.
log 2 8  3
log3 3  1 2
log 7 7  1
4.
log 9 1  0
log 6 11
 11
5.
6
6.
1
log 5
 2
25
What’s true about the (x, y) pairs and graphs of inverse functions?
x
f  x  2
x
x f  x   log2 x
1
–3
1/8
1/8
–3
–2
1/4
1/4
–2
–1
1/2
1/2
–1
0
1
1
0
1
2
2
1
2
4
4
2
3
8
8
3
Now, let’s plot these points and discuss the graphs…
Common Logarithms
Common Logarithm – logarithm with a base of 10
(very commonly used because of our base 10 number system!)
For common logarithms, we can drop the subscript:
y  log x
if and only if
10  x
y
Basic Properties of
Common Logarithms
Let x and y be real numbers with x > 0.
1.
2.
3.
4.
log1  0
10  1
1
log10  1 because 10  10
y
y
y
log10  y because 10  10
10
log x
because
x
because
0
log x  log x
More Evaluating Logarithms
Evaluate each of the following.
1.
log100  2
1
3. log
 3
1000
2.
 0.367
2
4. log 5 100 
5
log10
0.367
Note: The LOG key on your calculator refers
to the common logarithm…
Using Your Calculator
Use a calculator to evaluate the logarithmic expression if it is
defined, and check your result by evaluating the corresponding
exponential expression.
1.537...
b/c
1.
log 34.5  1.537...
10
 34.5
0.366...
 0.43
2.
log 0.43  0.366...
3.
log  3 is undefined  can you explain why ?
b/c
10
Solving Simple Logarithmic
Equations
Solve the given equations by changing to exponential form.
1.
log x  3
Exp. Form:
2.
log 2 x  5
Exp. Form:
x  10
 x  1000
3
x2
 x  32
5
What is the definition of the natural base???
 1
e  lim  1  
x 
 x
x
Natural Logarithm – a logarithm with base e
Notation: ln
That is,
loge x  ln x
Back to our inverse relationship:
y  ln x
if and only if
e x
y
Let x and y be real numbers with x > 0.
e 1
1
2. ln e  1 because e  e
y
y
y
3. ln e  y because e  e
ln x
4. e
 x because ln x  ln x
1.
ln1  0
because
0
Evaluate each of the following without a calculator.
1.
2.
ln e  1 2
ln e  5
5
3.
e
ln 4
4
Note: The LN key on your calculator refers
to the natural logarithm…
Use a calculator to evaluate the given logarithmic expressions,
if they are defined, and check your result by evaluating the
corresponding exponential expression.
1.
ln 23.5  3.157... because e3.157...  23.5
2.
ln  5.43 is undefined!!!
3.
ln 0.48  0.733...
Why???
because
e
0.733...
 0.48
Solve each of the given equations by changing them to
exponential form.
1.
log x  5
2.
x = 100,000
3.
log x  2
x=
1
100
= 0.01
ln x  1
x=
4.
1
e
= 0.368…
ln x  2.5
x=e
2.5
= 12.182…