# Digging Laws of Logarithm Deeper

In order to evaluate logarithmic expression we need to know the basic laws of logarithm. This topic is still not that easy because it’s very confusing. On how to deal with this easily, practice is the key. This website is dedicated to that.

Basic Laws:

**I. Addition-Multiplication Law**

Example:

Think of this,

This is the common misconception about this addition law.

**II. Subtraction-Division Law**

Example:

Think of this,

**III. Power Law**

Example:

**Worked Problem 1:**

Given , , . Find the value $latex

\log(84)$

[toggle title=”Solution”]

We need to express in terms of the given quantities.

$latex \log(84)=\log(4\cdot 3\cdot 7)= \log(2^2\cdot 3\cdot 7)=\log(2^2)+\log(3)+\log(7)

\to 2\log(2)+\log(3)+\log(7)=2q+p+r$ [/toggle]

**Worked Problem 2**: 14^{th} Philippine Math Olympiad Qualifying Round

Let and . Express in terms of *r* and *s*.

(a)2r+s (b) 2r+3s (c) r+2s (d) 3r+2s

[toggle title=”Solution”]

Using power rule,

Work first if how many we can get out of

. The answer is letter (b). [/toggle]

**Worked Problem 3: **2007 MMC Elimination

Evaluate and simplify:

[toggle title=”Solution”]

Take note that .

The easiest approach to this problem is to express and to the base 2 since the base of logarithm is 2.

Going back to the original expression,

[/toggle]

**Worked Problem 4: **2013 MMC Elimination

If , and , express in terms of p,q, and r.

[toggle title=”Solution”]

Take note for this identity taken from change of base.

take note of this identity and we will use this later.

$latex \log50+2\log70-\log_27

=\log(5^2\cdot2)+2\log(2\cdot 5\cdot 7)-\log_27

=2\log5+\log2+2(\log2+\log5+\log7)-\displaystyle\frac{\log7}{\log2}

=2\log5+\log2+2\log2+2\log5+2\log7)-\displaystyle\frac{\log7}{\log2}

=4\log5+3\log2+2\log7-\displaystyle\frac{\log7}{\log2}

=4q+3p+2r-\displaystyle\frac{r}{p}$ [/toggle]

Don’t miss the next topic about change of base property of logarithm.

### Dan

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