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Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
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Solving Equations with Log Terms on Each Side

When solving equations with two or more logs it is often necessary to use the properties of logarithms.

Log of a Product Property:

Log of a Quotient Property:

Log of a Power Property:

logb u · v = logb u + logb v

logb = logb u - logb v

logb un = n · logbu


Here is another useful property. It says that if the logs of two quantities are equal, then the quantities are equal.


Property — Logarithmic Equality

If logb x = logb y, then x = y.

Here, x > 0, y > 0, b > 0 and b 1.


In the following examples, we will first use the properties of logarithms to rewrite each equation so there is one log on each side. Then, we will use the Principal of Logarithmic Equality.


Example 1

Solve: log 4 + log x = log 14


Use the Log of a Product Property.

Use the Principal of Logarithmic Equality.

Divide both sides by 4.

log 4 + log x

log 4x



= log 14

= log 14

= 14

= 3.5

 So, the solution is x = 3.5.



To check the solution of log 4 + log x = log 14, substitute 3.5 for x and simplify.





 log 4 + log x

log 4 + log 3.5

0.602 + 0.544


= log 14

= log 14 ?

= 1.146 ?

= 1.146 = Yes

Because we must round the values of the logs, in some cases the two sides of the equation will not simplify to exactly the same result. However, the results should be very close.


Example 2

Solve: log (x + 4) - log 6 = log (x - 11).

Solution log (x + 4) - log 6 = log (x - 11)
Use the Log of a Quotient Property. = log (x - 11)
Use the Principal of Logarithmic Equality. = x - 11
Multiply both sides by 6.

Subtract x from both sides.

Add 66 to both sides.

Divide both sides by 5.

 x + 4




= 6x - 66

= 5x - 66

= 5x

= x

So, the solution is x = 14. We leave the check to you.
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