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Lecture #19 of Math 110, focusing on logarithmic functions, logarithmic equations, and their properties. Students will learn about the Change-of-Base Formula, the product rule, quotient rule, power rule, inverse property, expanding and condensing logarithmic expressions, and solving the simplest logarithmic equations. The lecture includes examples and problems to help students understand these concepts.
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CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.
Change-of-Base Formula.
For any logarithmic bases a and b , and any
positive number M ,
log log log
a b a
b
Problem #1.
Use your calculator to find the following logarithms.
Show your work with Change-of-Base Formula.
a) log 10 2 b) 1
3
log 9 c) log 11 7
Using the Change-of-Base Formula, we can graph
Logarithmic Functions with an arbitrary base.
Example :
2
2
ln log ln 2
log log log 2
x x
x x
y =log 2 x
CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.
Properties of Logarithms.
If b, M, and N are positive real numbers, b ≠ 1 , p , x are real
numbers, then
M (^) bN N
= − quotient rule
p b M^ =^ p bM^ power rule
log
b , 0
x b
x
b x
b x x
This property is the base for solving Logarithmic
Properties 1-3 may be used for Expanding and Condensing
Logarithmic expressions.
Expanding and Condensing Logarithmic expressions.
CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.
Given: lo g b x = a , b > 0 , b ≠ 1 , a is any real number.
According the definition of the logarithm this equation is
equivalent to
a x = b.
log (^) b M = log bN , then M = N.
Remember, check is part of solution for
Logarithmic Equations.
Problem #4. Solve the following Logarithmic Equations.
a) log 2 x = 5
2 log x − x =log 6
2
log x + 4 = − 3
CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.