Beginning with an In-Class Activity
Section 6-5Properties of Logarithms
Logarithm of 1
Logarithm of 1
For any base b,
logb 1 = 0
Logarithm of 1
For any base b,
logb 1 = 0
Why?
Logarithm of 1
For any base b,
logb 1 = 0
Why?
logb 1 = 0
Logarithm of 1
For any base b,
logb 1 = 0
Why?
logb 1 = 0
b0
Logarithm of 1
For any base b,
logb 1 = 0
Why?
logb 1 = 0
b0 = 1
Logarithm of a Product
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb y
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
xy = bmibn
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
xy = bmibn
= bm+n
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
xy = bmibn
= bm+n
logb xy = m+n
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
xy = bmibn
= bm+n
logb xy = m+n
logb x = m logb y = n
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
xy = bmibn
= bm+n
logb xy = m+n
logb x = m logb y = nSo,
Logarithm of a ProductFor any base b and for any positive real numbers x and y,
logb(xy) = logb x + logb yWhy?
logb x = m logb y = n
bm = x b
n = y
xy = bmibn
= bm+n
logb xy = m+n
logb x = m logb y = n
logb(xy) = logb x + logb ySo,
Logarithm of a Quotient
Logarithm of a Quotient
For any base b and for any positive real numbers x and y,
logb
xy( ) = logb x − logb y
Logarithm of a Quotient
For any base b and for any positive real numbers x and y,
logb
xy( ) = logb x − logb y
Why?
Logarithm of a Quotient
For any base b and for any positive real numbers x and y,
logb
xy( ) = logb x − logb y
Why?
Well, now, that’s a homework problem!
Logarithm of a Power
Logarithm of a Power
For any base b, and real positive real number x, and any real number p,
logb x p = p logb x
Logarithm of a Power
For any base b, and real positive real number x, and any real number p,
logb x p = p logb x
Why?
Logarithm of a Power
For any base b, and real positive real number x, and any real number p,
logb x p = p logb x
Why?
It’s another homework problem!
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
log6 2+ log6 3 = 1
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
= log5
2008( )
log6 2+ log6 3 = 1
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
= log5
2008( )
= log5(25)
log6 2+ log6 3 = 1
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
= log5
2008( )
= log5(25)
5x = 25
log6 2+ log6 3 = 1
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
= log5
2008( )
= log5(25)
5x = 25
x = 2
log6 2+ log6 3 = 1
Example 1Simplify without a calculator. In other words, let’s use
what we know about logarithms!
a. log6 2+ log6 3 b. log5 200− log5 8
= log6(2i3)
= log6(6)
6x = 6
x = 1
= log5
2008( )
= log5(25)
5x = 25
x = 2
log6 2+ log6 3 = 1 log5 200− log5 8 = 2
Example 1
c. log85− log17 + 12
log400
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
= log5+ 12
log400
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
= log5+ 12
log400
= log5+ log40012
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
= log5+ 12
log400
= log5+ log40012
= log5+ log20
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
= log5+ 12
log400
= log5+ log40012
= log5+ log20
= log(5i20)
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
= log5+ 12
log400
= log5+ log40012
= log5+ log20
= log(5i20)
= log100
Example 1
c. log85− log17 + 12
log400
= log 85
17( )+ 12
log400
= log5+ 12
log400
= log5+ log40012
= log5+ log20
= log(5i20)
= log100 = 2
Example 2Rewrite without logarithms.
a. ln x = 1
3lna b. log x = loga − 4 logb
Example 2Rewrite without logarithms.
a. ln x = 1
3lna b. log x = loga − 4 logb
ln x = lna13
Example 2Rewrite without logarithms.
a. ln x = 1
3lna b. log x = loga − 4 logb
ln x = lna13
x = a13
Example 2Rewrite without logarithms.
a. ln x = 1
3lna b. log x = loga − 4 logb
ln x = lna13
x = a13
log x = loga − logb4
Example 2Rewrite without logarithms.
a. ln x = 1
3lna b. log x = loga − 4 logb
ln x = lna13
x = a13
log x = loga − logb4
log x = log a
b4( )
Example 2Rewrite without logarithms.
a. ln x = 1
3lna b. log x = loga − 4 logb
ln x = lna13
x = a13
log x = loga − logb4
log x = log a
b4( ) x = a
b4
Example 3Write an equation without exponents equivalent to the
exponential equation A = Pert.
Example 3Write an equation without exponents equivalent to the
exponential equation A = Pert.
A = Pert
Example 3Write an equation without exponents equivalent to the
exponential equation A = Pert.
A = Pert
P P
Example 3Write an equation without exponents equivalent to the
exponential equation A = Pert.
A = Pert
P P
AP= ert
Example 3Write an equation without exponents equivalent to the
exponential equation A = Pert.
A = Pert
P P
AP= ert
lnAP= lnert
Example 3Write an equation without exponents equivalent to the
exponential equation A = Pert.
A = Pert
P P
AP= ert
lnAP= lnert
ln A− lnP = rt
Example 4Which number is larger, 400500 or 500400?
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500 = 500log400
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500 = 500log400 ≈ 1301.029996
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500 = 500log400 ≈ 1301.029996
log y = log500400
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500 = 500log400 ≈ 1301.029996
log y = log500400 = 400log500
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500 = 500log400 ≈ 1301.029996
log y = log500400 = 400log500 ≈ 1079.588002
Example 4Which number is larger, 400500 or 500400?
Let x = 400500 and y = 500400.
log x = log400500 = 500log400 ≈ 1301.029996
log y = log500400 = 400log500 ≈ 1079.588002
So 400500 is larger.
Homework
Homework
p. 401 #1-25
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