A Guide to Common and Natural Logarithms with Natural Base

First, simplify the expressions on the left. They can then be paired with their corresponding expressions on the right.

Expression $1$

To simplify this expression, the Quotient Property of Logarithms and the definition of a common logarithm will be used.

lo g 10 x - lo g x

$lo g (m) - lo g (n) = lo g (\frac{m}{n})$

lo g \frac{1 0 x}{x}

CancelCommonFac

Cancel out common factors

lo g \frac{1 0 x}{x}

SimpQuot

Simplify quotient

lo g 10

$lo g (10) = 1$

1

The expression

lo g 10 x - lo g x

should be paired with

1 .

Expression $2$

To simplify this expression, the Power Property of Logarithms, the Product Property of Logarithms, and the definition of a common logarithm will be used.

2 lo g x + lo g 10 x

$m \cdot lo g (a) = lo g (a^{m})$

lo g x^{2} + lo g 10 x

$lo g (m n) = lo g (m) + lo g (n)$

lo g x^{2} + lo g 10 + lo g x

$lo g (10) = 1$

lo g x^{2} + 1 + lo g x

CommutativePropAdd

Commutative Property of Addition

1 + lo g x + lo g x^{2}

$lo g (m) + lo g (n) = lo g (m n)$

1 + lo g (x \cdot x^{2})

MultBasePow

$a \cdot a^{m} = a^{1 + m}$

1 + lo g x^{3}

The expression

2 lo g x + lo g 10 x

should be paired with

1 + lo g x^{3} .

Expression $3$

To simplify the third expression, the Product Property of Logarithms, the Quotient Property of Logarithms, a logarithm identity, and the definition of a common logarithm will be used.

lo g 0.1 x

$lo g (m n) = lo g (m) + lo g (n)$

lo g 0.1 + lo g x

NumberToFrac

$a = \frac{1 0 \cdot a}{1 0}$

lo g \frac{1 0 \cdot 0 . 1}{1 0} + lo g x

Multiply

lo g \frac{1}{1 0} + lo g x

$lo g (\frac{m}{n}) = lo g (m) - lo g (n)$

lo g 1 - lo g 10 + lo g x

$lo g (1) = 0$

0 - lo g 10 + lo g x

$lo g (10) = 1$

0 - 1 + lo g x

SubTerm

Subtract term

- 1 + lo g x

CommutativePropAdd

Commutative Property of Addition

lo g x - 1

The expression

lo g 0.1 x

should be paired with

lo g x - 1 .

Expression $4$

To simplify the second to last expression, the Quotient Property of Logarithms, the Power Property of Logarithms, and the definition of a common logarithm will be used.

lo g \frac{1 0 0}{x} + lo g x^{2}

$lo g (\frac{m}{n}) = lo g (m) - lo g (n)$

lo g 100 - lo g x + lo g x^{2}

WritePow

Write as a power

lo g 1 0^{2} - lo g x + lo g x^{2}

$lo g (a^{m}) = m \cdot lo g (a)$

2 lo g 10 - lo g x + 2 lo g x

$lo g (10) = 1$

2 (1) - lo g x + 2 lo g x

IdPropMult

Identity Property of Multiplication

2 - lo g x + 2 lo g x

AddTerms

Add terms

2 + lo g x

The expression

lo g \frac{1 0 0}{x} + lo g x^{2}

should be paired with

2 + lo g x .

Expression $5$

Finally, to simplify the last expression, the Product Property of Logarithms and the Power Property of Logarithms will be used.

lo g x + lo g x^{2} - 3 lo g x

$lo g (m) + lo g (n) = lo g (m n)$

lo g (x \cdot x^{2}) - 3 lo g x

MultBasePow

$a \cdot a^{m} = a^{1 + m}$

lo g x^{3} - 3 lo g x

$m \cdot lo g (a) = lo g (a^{m})$

lo g x^{3} - lo g x^{3}

SubTerm

Subtract term

0

The expression

lo g x + lo g x^{2} - 3 lo g x

should be paired with

0 .

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Catch-Up and Review

Mathematical Operations

Common Logarithm

Evaluating Common Logarithms

Earthquakes and Logarithms

Hint

Solution

Using the Properties of Common Logarithms

Hint

Solution

Expression $1$

Expression $2$

Expression $3$

Expression $4$

Expression $5$

Calculating the Value of an Expression for Bigger Values of $n$

The Natural Base

The Number $e$

The Natural Logarithm

Evaluating Natural Logarithms

Using the Properties of Natural Logarithms

Hint

Solution

Expression $1$

Expression $2$

Expression $3$

Expression $4$

Expression $5$

Change of Base Formula

Change of Base Formula

Proof

Using the Change of Base Formula

Hint

Solution

Leonhard Euler and His Contributions

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Catch-Up and Review

Hint

Solution

Hint

Solution

Expression 1

Expression 2

Expression 3

Expression 4

Expression 5

Hint

Solution

Expression 1

Expression 2

Expression 3

Expression 4

Expression 5

Proof

Hint

Solution

Expression $1$

Expression $2$

Expression $3$

Expression $4$

Expression $5$

Expression $1$

Expression $2$

Expression $3$

Expression $4$

Expression $5$