Logarithmic Theorems 1

Hey everyone! Okay, so today we learned about the laws of logarithms.

Laws of Logarithms:

Multiplication Law

Division Law

Exponent Law

Here are the Logarithmic Expansion Steps:

1) Roots become fractional exponents.

2) Division Law- Division becomes subtraction.

3) Multiplication Law- Multiplication becomes addition.

4) Exponent Law- Exponents becomes coefficients (move to the front)

Here are a couple of examples of expansion of some logarithms below.

1) ex: log₅12⁴/11²

Since there is division inside the log, the division would become subtraction.

log₅12⁴/11²= log₅12⁴- log₅11²

The exponents, 4 and 2 would then become coefficients and move to the front and the final answer would be:

log₅12⁴/11²= 4log₅12 - 2log₅11

2) ex: log₆(uv⁶)⁵

Since there is multiplication inside the log, the multiplication would become addition.

log₆(uv⁶)⁵= log₆u⁵ + log₆(v⁶)⁵

The exponents, 5 and 6x5=30 would be the coefficients and move to the front and the answer would be:

log₆(uv⁶)⁵= 5log₆u + 30log₆v

3) ex: log₆(5√12⋅11)

Since there is multiplication inside the log, it would become addition. x½= the square root of x so:

log₆(5√12⋅11)= log₆5 + log₆(12)½ + log₆(11)½

The exponent 1/2 would become a coefficient and move to the front and the answer would be:

log₆(5√12⋅11)= log₆5 + 1/2log₆12 + 1/2log₆11

One more thing. When the base is not provided, assume that the base is 10 like the example shown below.

log4x²y³/√z= log₁₀4x²y³ - log₁₀√z

log4x²y³/√z= log₁₀4 + log₁₀x² + log₁₀y³ - log₁₀(z)½

log4x²y³/√z= log₁₀4 + 2log₁₀x + 3log₁₀y - 1/2log₁₀z

I think that this is it and I hope that I covered everything, just follow and remember the logarithmic expansion steps and everything will come naturally. See you all tomorrow in class!