Algebra Tutorials!  
Wednesday 17th of April
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Logarithmic Functions

Recall that all exponential functions of the form y = bx where a > 0 and a 1 are one-to-one. This means that they have inverses. Their inverses satisfy the equation: x = by (b > 0, b 1)

These inverse functions, when solved for y, are called Logarithmic Functions.

The , where b > 0, b 1, is denoted by y = log b x and is defined by:

y = log b x iff x = by

In words, this definition says: "y is the power (exponent) to which b must be raised in order to get x".

What you need to remember is that a logarithm is an exponent.

In the above definition, y is the logarithm and y is the exponent.


1) 25 = 32  is equivalent to 5 = log 2 32
2) 102 = 100  is equivalent to 2 = log 10 100
3)  is equivalent to
4) 61 = 6  is equivalent to 1 = log 6 6
5) 70 = 1  is equivalent to 0 = log 7 1


To find the exact value of a logarithm, we write the logarithm in exponential notation and use this property of exponential functions:

ONE-TO-ONE PROPERTY OF EXPONENTS: If au = av, then u = v


Domain of a Logarithmic Function

Remember that an exponential function has domain, D = the set of real numbers and range, R = the set of positive real numbers. Therefore, since they are inverses, the domain of a logarithmic function must be D = the set of positive real numbers, and its range must be R = the set of real numbers.

This means that the argument of a logarithm must always be positive ( greater than 0). 


Graphs of Logarithmic Functions

The functions y = bx and y = log b x are inverses of each other. Recall what the graph of y = bx looks like. Then try to picture the graph of its reflection through the line y = x.


All graphs of the form y = log b x, when b > 1, will have the same basic shape as this one.


Facts About the Graphs of Logarithmic Functions f(x) = log b x

1) The domain is the set of positive real numbers; the range is all real numbers.

2) The x-intercept is 1; there is no y-intercept.

3) The y-axis (x = 0) is the vertical asymptote.

4) f(x) = log b x, b >1, is an increasing function and is one-to-one.

f(x) = log b x, 0 < b < 1, is a decreasing function and is one-to-one.

5) The graph contains the points (1, 0) and (b, 1).

6) The graph is smooth and continuous with no corners or gaps.


COMMON LOGARITHMS are logarithms with base 10. The common logarithmic function would be y = log 10 x, except that the 10 is not written.

y = log x ↔ 10y = x

In words, "y is the exponent to which 10 must be raised in order to get x".

log 100 = 2, since 102 = 100

log 1000 = 3, since 103 = 1000

log (0.1) = -1, since 10-1 = 0.1


The Natural Logarithmic Function

Recall y = ex, the exponential function. Its inverse would be a logarithmic function with base e, or y = log e x. However, because of the significance of the number e, this logarithmic function has a special oe e name, the natural logarithmic function, and it is written ln x. 

Definition of The Natural Logarithmic Function:

y = ln x iff x = ey

In words: "y is the exponent to which e must be raised in order to get x".

Since e is a number greater than 1, the graph of the natural logarithmic function is shaped like the graph of y = log 2 x:


If we need to find logarithmic values in bases other than 10 and e, we can use a formula called the:

CHANGE-OF-BASE FORMULA: If b > 0, b 1, c > 0, c 1and x > 0, then

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