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5.05 Logarithm laws

Lesson

In the same way that there are index laws which allow us to simplify exponential expressions, there are logarithm laws that allow us to simplify logarithmic expressions. In fact, each logarithm law is a consequence of an index law.

First, consider the definition that if $y=B^x$y=Bx then $x=\log_By$x=logBy. It follows that,

$\log_BB^x=x$logBBx=x

Substituting $x=0$x=0 and $x=1$x=1 gives the following special cases:

$\log_B1=0$logB1=0

$\log_BB=1$logBB=1

For the following proofs, we will let $p=B^m$p=Bm and $q=B^n$q=Bn so that $\log_Bp=m$logBp=m and $\log_Bq=n$logBq=n. Note that for any $B\ne0$B0 there will be some values of $m,n,p,$m,n,p, and $q$q which makes these equations true.

$B^m\times B^n$Bm×Bn $=$= $B^{m+n}$Bm+n

From the index law

$\log_BB^m\times B^n$logBBm×Bn $=$= $\log_BB^{m+n}$logBBm+n

Taking the logarithm base $B$B from both sides

$\log_BB^m\times B^n$logBBm×Bn $=$= $m+n$m+n

From the definition of logarithms

$\log_Bpq$logBpq $=$= $\log_Bp+\log_Bq$logBp+logBq

Substituting values of $m,n,p,$m,n,p, and $q$q

 

$\frac{B^m}{B^n}$BmBn $=$= $B^{m-n}$Bmn

From the index law

$\log_B\frac{B^m}{B^n}$logBBmBn $=$= $\log_BB^{m-n}$logBBmn

Taking the logarithm base $B$B from both sides

$\log_B\frac{B^m}{B^n}$logBBmBn $=$= $m-n$mn

From the definition of logarithms

$\log_B\frac{p}{q}$logBpq $=$= $\log_Bp-\log_Bq$logBplogBq

Substituting values of $m,n,p,$m,n,p, and $q$q

 

$\left(B^m\right)^n$(Bm)n $=$= $B^{mn}$Bmn

From the index law

$\log_B\left(B^m\right)^n$logB(Bm)n $=$= $\log_BB^{mn}$logBBmn

Taking the logarithm base $B$B from both sides

$\log_B\left(B^m\right)^n$logB(Bm)n $=$= $mn$mn

From the definition of logarithms

$\log_Bp^n$logBpn $=$= $n\log_Bp$nlogBp

Substituting values of $m$m and $p$p

Summary

The definition of logarithms and the index laws give us the following results:

$\log_BB^x=x$logBBx=x

$\log_B1=0$logB1=0

$\log_BB=1$logBB=1

$\log_Bpq=\log_Bp+\log_Bq$logBpq=logBp+logBq

$\log_B\frac{p}{q}=\log_Bp-\log_Bq$logBpq=logBplogBq

$\log_Bp^n=n\log_Bp$logBpn=nlogBp

Practice questions

Question 1

Simplify $\frac{\log_449}{\log_47}$log449log47.

Question 2

Express $\log2x+\log50y$log2x+log50y in simplest form.

Question 3

Use the properties of logarithms to rewrite the expression $\log_8\left(x^6\right)$log8(x6).

Write your answer without any powers.

Outcomes

VCMNA356 (10a)

Use the definition of a logarithm to establish and apply the laws of logarithms and investigate logarithmic scales in measurement

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