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What Are the Logarithm Rules?

As you can see below, all the logarithmic rules for the common logarithm log x and the natural logarithm ln x are the same:

Common logarithm:

10log a = a

Natural logarithm:

eln a = a

This is true for any logarithm regardless of its base.

Rule

The Logarithmic Rules for the Common Logarithm

The first logarithmic rule:

log (ax) = x log a

The second logarithmic rule:

log (a b) = log a + log b

The third logarithmic rule:

log (a b) = log a log b

Rule

The Logarithmic Rules for the Natural Logarithm

The first logarithmic rule

ln (ax) = x ln a

The second logarithmic rule

ln (a b) = ln a + ln b

The third logarithmic rule

ln (a b) = ln a ln b

Example 1

Simplify log a2 + log b2 2 log a

log a2 + log b2 2 log a = 2 log a + 2 log b 2 log a = 2 log b

log a2 + log b2 2 log a = 2 log a + 2 log b 2 log a = 2 log b

Example 2

Simplify log ab + log b2 log a2b

log ab + log b2 log a2b = log a + log b + 2 log b (log a2 + log b) = log a + 3 log b log a2 log b = log a + 2 log b 2 log a = log a + 2 log b

log ab + log b2 log a2b = log a + log b + 2 log b (log a2 + log b) = log a + 3 log b log a2 log b = log a + 2 log b 2 log a = log a + 2 log b

Example 3

Simplify log a b log 2a b3

log a b log 2a b3 = log a log b (log 2a log b3) = log a log b (log 2 + log a 3 log b) = log a log b log 2 log a + 3 log b = 2 log b log 2

log a b log 2a b3 = log a log b (log 2a log b3) = log a log b (log 2 + log a 3 log b) = log a log b log 2 log a + 3 log b = 2 log b log 2

Example 4

Simplify log 2x + log 2 log 2 x2 + log 10

log 2x + log 2 log 2 x2 + log 10 = log 2 + log x + log 2 (log 2 log x2) + 1 = 2 log 2 + log x log 2 + log x2 + 1 = log 2 + log x + 2 log x + 1 = log 2 + 3 log x + 1

log 2x + log 2 log 2 x2 + log 10 = log 2 + log x + log 2 (log 2 log x2) + 1 = 2 log 2 + log x log 2 + log x2 + 1 = log 2 + log x + 2 log x + 1 = log 2 + 3 log x + 1

Example 5

Use the logarithmic rules to simplify ln 2x ln (x 2) 4 ln x

= ln 2x ln (x 2 ) 4 ln x = ln 2 + ln x (ln x ln 2) 4 ln x = ln 2 + ln x ln x + ln 2 4 ln x = 2 ln 2 4 ln x

ln 2x ln (x 2 ) 4 ln x = ln 2 + ln x (ln x ln 2) 4 ln x = ln 2 + ln x ln x + ln 2 4 ln x = 2 ln 2 4 ln x

Example 6

Use the logarithmic rules to simplify ln 2x3 ln (3x 2 ) + ln (3x)2

= ln 2x3 ln (3x 2 ) + ln (3x)2 = ln 2 + ln x3 (ln 3x ln 2) + ln 32x2 = ln 2 + 3 ln x (ln 3 + ln x ln 2) + ln 32 + ln x2 = ln 2 + 3 ln x ln 3 ln x + ln 2 + 2 ln 3 + 2 ln x = 2 ln 2 + 4 ln x + ln 3

ln 2x3 ln (3x 2 ) + ln (3x)2 = ln 2 + ln x3 (ln 3x ln 2) + ln 32x2 = ln 2 + 3 ln x (ln 3 + ln x ln 2) + ln 32 + ln x2 = ln 2 + 3 ln x ln 3 ln x + ln 2 + 2 ln 3 + 2 ln x = 2 ln 2 + 4 ln x + ln 3

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