Everything Maths Grade 12

CHAPTER 2. LOGARITHMS 2.

The right hand side:

log 10− log 100 = 1− 2

=− 1

Both sides are equal. Therefore, log( 10010 ) = log 10− log 100.

Activity: Logarithm Law 4:loga

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x

y

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=loga(x)−loga(y)

Write as separate logs:

1. log 2 (^85 )


2. log 8 (^1003 )


3. log 16 (xy)


4. logz(^2 y)


5. logx(y 2 )

2.9 Logarithm Law 5: loga(xb) = b loga(x)

EMCJ

Once again, we need torelate x to the base a. So, we let x = am. Then,

loga(xb) = loga((am)b)

= loga(am. b) (Exponential Law in Equation (Grade 10))

But, m = loga(x) (Assumption that x = am)

∴ loga(xb) = loga(ab .loga(x))

= b. loga(x) (Definition of logarithm in Equation 2.1)

For example, we can show that log 2 (5^3 ) = 3 log 2 (5).

log 2 (5^3 ) = log 2 (5. 5. 5)

= log 2 5 + log 2 5 + log 2 5 (∵ loga(x.y) = loga(am.an))

= 3 log 25

Therefore, log 2 (5^3 ) = 3 log 2 (5).

Activity: Logarithm Law 5:loga(xb)=bloga(x)

Simplify the following:

1. log 2 (8^4 )


2. log 8 (10^10 )


3. log 16 (xy)


4. logz(yx)


5. logx(y^2 x)