College Algebra

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Previous Lessons
Open Chapter Ch. 1: Graphs, Functions, and Models
Lesson #1 Graphing
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Lesson #2 Graphing Functions
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Lesson #3 Slope of Linear Functions
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Lesson #4 Linear Equations and Modeling
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Lesson #5 Linear Equations, Functions, Zeros and Applications
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Lesson #6 Linear Inequalities
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Open Chapter Ch. 2: More on Functions
Lesson #7 Analyzing Functions
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Exam Exam 1
Lesson #8 Algebra for Functions
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Lesson #9 The Composition of Functions
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Lesson #10 Symmetry
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Lesson #11 Transformations
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Lesson #12 Equations of Variation
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Open Chapter Ch. 3: Quadratic Functions and Equations; Inequalities
Lesson #13 Complex Numbers
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Lesson #14 Quadratic Equations, Functions, Zeros and Models
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Lesson #15 Analyzing Graphs of Quadratic Functions
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Lesson #16 Rational and Radical Equations
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Exam Midterm Exam
Lesson #17 Absolute Value Equations and Inequalities
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Open Chapter Ch. 4: Polynomial Functions and Rational Functions
Lesson #18 Polynomial Functions and Models
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Lesson #19 Graphing Polynomial Functions
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Lesson #20 Polynomial Division
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Lesson #21 The Zeros of Polynomial Functions
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Lesson #22 Rational Functions
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Lesson #23 Polynomial and Rational Inequalities
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Open Chapter Ch. 5: Exponential Functions and Logarithmic Functions
Lesson #24 Inverse Functions
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Lesson #25 Exponential Functions and Graphs
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Lesson #26 Logarithmic Functions and Graphs
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Exam Exam 3
Lesson #27 Properties of Logarithmic Functions
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Lesson #28 Solving Exponential Equations and Logarithmic Equations
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Lesson #29 Applications and Models: Growth and Decay; Compound Interest
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Open Chapter Ch. 6: Systems of Equations
Lesson #30 Systems of Equations in Two Variables
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Lesson #31 Systems of Inequalities and Linear Programming
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Lesson #32 Nonlinear Systems of Equations and Inequalities
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Lesson #33 Sequences and Series
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Exam Final Exam

Assignments:

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Lesson Objectives:

- The Product Rule
- The Power Rule
- The Quotient Rule
- The Logarithm of a Base to a Power
- A Base to a Logarithmic Power


The Product Rule:

The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.

Let m and n be any two numbers, and x and y their logarithms. Then by definition, `10^x = m`, and `x = log m`. And `10^y = n`, and `y = log n`.

Multiplying `10^x*10^y = m*n`, or `10^(x+y) = m*n`, because we can add their exponents. So the `log(m*n) = x + y`, because we converted this to log form.

And now substitute in your x and your y-values, `log m` and `log n`, so that we get the product rule, which is `log(m*n) = log m + log n`.



Express as a single logarithm and if possible, simplify.

`log_a50+log_a2`. So because of the Product Rule, we can rewrite his as `log_a(50*2)`, which is `log_a(100)`.



The Power Rule:

The logarithm of any power of a number is equal to the logarithm of the quantity multiplied by the exponent of the power.

Let m be any number, and x its logarithm. Then, by definition, `10^x = m`, and `log m = x`.

So if we raise both members to a power denoted by p, then we get `10^(p*x) = m^p`, or if we convert this to log form, we have `log m^p = p*x`. And then substitute `log m` in for x, and we get `p*log m`.

So the Power Rule is `log m^p = p*log m`.



Express as a product.

`log_ak^(-3)`. So by the Power Rule, we can simplify this to `-3*log_ak`.



The Quotient Rule:

The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.

Let `m` and `n` be any two numbers, and `x` and `y` their logarithms. Then by definition, `10^x = m`, and `log m = x`. `10^y = n`, and `log n = y`.

So if we divide `(10^x)/(10^y)`, this is equal to `m/n`, or `10^(x-y) = m/n`. And if we convert this to log form, this is the same as `log(m/n) = x - y`.

Now substitute the values of `x` and `y`, `log m` and `log n`, and you have `log(m/n) = log m - log n`, which is the Quotient Rule.



Express as a single logarithm and simplify:

`log(1,000,000)-log(1,000)`. By the Quotient Rule, we can rewrite this as `log(1,000,000/1,000)`, and then we can simplify by dividing the numerator and the denominator by 1,000. So we have `log 1,000`, which is the same as `log_(10)1000`, or `log_(10)10^3`, and by the power rule, this is the same as `3*log_(10)10`. `log_(10)10` is just 1, so this is the same as 3.



The Logarithm of a Base to a Power:

`log_aa^x = x` for any base, a, and any real number, x.

A Base to a Logarithmic Power:

`""_alog_ax = x` for any base, a, and any real number, x.



Simplify.

`""_5log5(3x)`. This can be simplified as 3x.