Theory of Logarithms: Laws of Logarithms and application of Logarithmic equations and indices

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Welcome to class! 

In today’s class, we will be talking about the theory of logarithms. Enjoy the class!

Logarithm Theory

Rules of logarithms

1) Product Rule:

The logarithm of a product is the sum of the logarithms of the factors.

log_a xy = log_a x + log_a y

2) Quotient Rule:

The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator

log_a (x/y) = log_a x – log_a y

3) Power Rule:

log_a xⁿ = nlog_a x

4) Change of Base Rule:

log_a b = log_c b / log_c a

log_a b = 1 / log_b a

where x and y are positive, and a > 0, a ≠ 1

Example:

Express log ₂ 4 + log ₂ 5 as a single logarithm

Solution

log ₂ 4 + log ₂ 5 = log ₂ (4 × 5) = log ₂ 20

The properties of logarithms

1. log_a 1 = 0 since a⁰ = 1.
2. log_a a = 1 since a¹ = a.
3. log_a a^x = x since a^x = a^x.

Product property: The log of a product equals the sum of the logs.
Quotient Property: The log of a quotient equals the difference of the logs.
Power Property: The log of power equals the product of the power and the log.

Logarithms applications

1. Population growth
2. Interest Rate Problems
3. Mortgage Problems
4. Radioactive Decay Problems
5. Earthquake Problems

Indices

Laws of Indices:

Fractional indices:

 

In our next class, we will be talking about Surds.  We hope you enjoyed the class.

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