LOGARITHMS MASTERY CENTER

Product Property

log_b(xy)=log_bx + log_by.

Example:log ₃ 6 y =log₃6 + log₃y

Power Property 

log_b(x^p)=plog_b(x)

Example:log ₂ 27 =log ₂ 3³=3log₂3

Quotient Property

log_b(x/y)=log_bx - log_by.

Example: log ₅ (3x+2)/(x-1) =log₅(3x+2) - log₅(x-1)

Power Property

log_bm^p= p log_b m

Example: log ₅ x³ =3 log ₅ x

Common Mistakes to avoid

These are the common mistakes to avoid when applying the properties of logarithms.

• The log of a sum is NOT the sum of the logs. The sum of the logs is the log of the product. The log of a sum cannot be simplified.
  log_a (x + y) ≠ log_a x + log_a y
• The log of a difference is NOT the difference of the logs. The difference of the logs is the log of the quotient. The log of a difference cannot be simplified.
  log_a (x - y) ≠ log_a x - log_a y
• An exponent on the log is NOT the coefficient of the log. Only when the argument is raised to a power can the exponent be turned into the coefficient. When the entire logarithm is raised to a power, then it can not be simplified.
  (log_a x)^r ≠ r * log_a x
• The log of a quotient is not the quotient of the logs. The quotient of the logs is from the change of base formula. The log of a quotient is the difference of the logs.
  log_a (x / y) ≠ ( log_a x ) / ( log_a y )

Properties of Logarithms

Logarithms Video Tutorials 

1. Review on Logarithms and e

2. Change of base

3. Natural Logarithm and e pt I  

4. Natural Logarithm and e pt II

5. Solving Logarithmic Equations pt I

6. Solving Logarithminc Equations pt II

7. Properties of Logarithms

8. Properties of Exponents pt I

9. Properties of Exponents pt II

10. Evaluating Logarithmic Expressions 

11. Exponential Growth and decay pt I

12. Exponential Growth and deacay pt II 

13. Solving Logarithmic Inequalities

14. Mixed Review on Logarithm and Exponents

15. Condensing Logarithms 

TESTS AND QUIZES

Interactive Practice Test

Logarithms Quiz » ProProfs Quiz Maker

Important Facts About Logarithms

Logarithms and exponents  are inverse functions. In solving equations  exponential equations, logarithms can be used to solve for the variable and vice versa. The inverse relationship between logarithms and exponents is illustrated in the statement below:

y=b^x ⇔ log_by=x

Logarithms base 10 are known as Common Logarithms and has many applications.
Logarithms base e are known as natural logarithm written with the symbol ln. Natural logarithms have multiple applications in mathematics, finance, and Engineering.  Logarithms were introduced by John Napier in the 17th century
Slide rules and logarithm tables were advanced computation devices in the past that were dependent on the properties of logarithms for their functionality.