1.9.4 Laws of logarithms

What you'll learn

What logarithms mean and when they are defined.
How to use the product, quotient and power laws.
How to expand one logarithm or combine several logarithms.
How the power law works with negative and fractional powers.

Before you start: what is a logarithm?

A logarithm is the inverse operation of raising a number to a power. It tells you which exponent produces a given number.

For example, because $2^3=8$ , it follows that $log_2 8=3$ .

Definition

Meaning of a logarithm

For a valid base $a$ ,

log⁡ax=b⟺ab=x.\log_a x=b \quad \Longleftrightarrow \quad a^b=x.

Here, $a$ is the base, $x$ is the argument and $b$ is the logarithm.

The base and argument must satisfy:

a>0,a≠1,x>0.a>0,\qquad a\neq 1,\qquad x>0.

In particular, you cannot take the logarithm of zero or a negative real number.

Example

Evaluating logarithms from powers

Evaluate $log_5 125$ and $log⁡4(12)\log_4 \left(\frac{1}{2}\right)$ .

Write 125 as a power of 5: $125=5^3$ . Therefore, $log_5 125=3$ .
Write $12\frac{1}{2}$ as a power of 4. Since $4−12=14=124^{-\frac12}=\frac{1}{\sqrt4}=\frac12$ , the required exponent is $−12-\frac12$ .
Therefore, $log⁡4(12)=−12\log_4 \left(\frac12\right)=-\frac12$ .

Why the laws work

The laws of logarithms follow from the laws of indices. For example, multiplying two powers with the same base means adding their exponents:

am⋅an=am+n.a^m\cdot a^n=a^{m+n}.

Logarithms reverse exponentiation, so multiplication inside a logarithm corresponds to addition outside it. Similarly, division corresponds to subtraction, and a power corresponds to multiplication.

Key Idea

Logarithms turn operations into simpler operations

For logarithms with the same base:

multiplication becomes addition;
division becomes subtraction;
a power becomes a multiplier.

The product law

The product law is

log⁡ax+log⁡ay≡log⁡a(xy).\log_a x+\log_a y\equiv\log_a(xy).

The symbol $≡\equiv$ means that the two expressions are identically equal whenever both sides are defined. The law works because if $x=a^m$ and $y=a^n$ , then

xy=a^m a^n=a^{m+n}.

Therefore, $log_a(xy)=m+n=\log_a x+\log_a y$ .

Example

Combining logarithms using the product law

Simplify $log_3 4+\log_3 9$ .

The logarithms have the same base and are being added, so apply the product law:
$log⁡34+log⁡39=log⁡3(4⋅9).\log_3 4+\log_3 9=\log_3(4\cdot 9).$
Calculate the product:
$log⁡3(4⋅9)=log⁡336.\log_3(4\cdot 9)=\log_3 36.$
Therefore, the combined expression is $log_3 36$ . It does not simplify to an integer because 36 is not an integer power of 3.

Common Mistake

Adding the arguments

The product law does not say $log_a x+\log_a y=\log_a(x+y)$ . Addition of logarithms produces a product of their arguments.

The quotient law

The quotient law is

log⁡ax−log⁡ay≡log⁡a(xy).\log_a x-\log_a y\equiv\log_a\left(\frac{x}{y}\right).

It comes from the index law

aman=am−n.\frac{a^m}{a^n}=a^{m-n}.

The order matters: the argument of the first logarithm becomes the numerator, and the argument of the subtracted logarithm becomes the denominator.

Example

Combining logarithms using the quotient law

Simplify $log_2 48-\log_2 3$ .

The logarithms have the same base and are being subtracted, so use the quotient law:
$log⁡248−log⁡23=log⁡2(483).\log_2 48-\log_2 3=\log_2\left(\frac{48}{3}\right).$
Simplify the quotient:
$log⁡2(483)=log⁡216.\log_2\left(\frac{48}{3}\right)=\log_2 16.$
Since $16=2^4$ , the expression equals 4.

Common Mistake

Reversing the quotient

In $log_a x-\log_a y$ , keep the same order: the result is $log⁡a(xy)\log_a\left(\frac{x}{y}\right)$ , not $log⁡a(yx)\log_a\left(\frac{y}{x}\right)$ .

The power law

The power law is

klog⁡ax≡log⁡a(xk).k\log_a x\equiv\log_a\left(x^k\right).

It allows a multiplier in front of a logarithm to become a power of its argument. The law can also be used in reverse:

log⁡a(xk)=klog⁡ax.\log_a\left(x^k\right)=k\log_a x.

This is especially useful when expanding logarithmic expressions.

Example

Using the power law in both directions

Rewrite $3\log_2 x$ as a single logarithm, and expand $log_5(y^7)$ .

Move the multiplier 3 into the first logarithm as a power:
$3\log_2 x=\log_2(x^3).$
Move the power 7 in the second expression outside as a multiplier:
$log_5(y^7)=7\log_5 y.$
These statements require the original logarithmic arguments to be positive, so $x > 0$ and $y^7>0$ .

Common Mistake

Multiplying the argument

The multiplier becomes an exponent, not a factor. Thus $k\log_a x=\log_a(x^k)$ , not $log_a(kx)$ .

Negative powers

The power law also applies when the multiplier is negative. In particular,

−log⁡ax=log⁡a(x−1)=log⁡a(1x).-\log_a x=\log_a\left(x^{-1}\right) =\log_a\left(\frac{1}{x}\right).

This is the case $k = - 1$ . It explains why changing the sign of a logarithm takes the reciprocal of its argument.

Example

Combining a logarithm with a negative coefficient

Write $log_a p-\log_a q$ as a single logarithm using the power and product laws.

Treat the subtraction as multiplication by negative one:
$log_a q=\log_a(q^{-1}).$
Use the product law to combine the logarithms:
$log_a p+\log_a(q^{-1})=\log_a(pq^{-1}).$
Since $q−1=1qq^{-1}=\frac1q$ , this becomes
$log⁡a(pq),\log_a\left(\frac{p}{q}\right),$
which agrees with the quotient law.

Fractional powers

A fractional power represents a root. For example,

x12=xandx−12=1x.x^{\frac12}=\sqrt{x} \qquad\text{and}\qquad x^{-\frac12}=\frac{1}{\sqrt{x}}.

Therefore, the power law gives

12log⁡ax=log⁡ax\frac12\log_a x=\log_a\sqrt{x}

and

−12log⁡ax=log⁡a(x−12)=log⁡a(1x).-\frac12\log_a x =\log_a\left(x^{-\frac12}\right) =\log_a\left(\frac{1}{\sqrt{x}}\right).

Example

Using a negative fractional coefficient

Rewrite $−12log⁡3x-\frac12\log_3 x$ as a single logarithm.

Use the power law to move the coefficient into the logarithm as an exponent:
$−12log⁡3x=log⁡3(x−12).-\frac12\log_3 x=\log_3\left(x^{-\frac12}\right).$
Interpret the exponent using the laws of indices:
$x−12=1x12=1x.x^{-\frac12}=\frac{1}{x^{\frac12}}=\frac{1}{\sqrt{x}}.$
Hence,
$−12log⁡3x=log⁡3(1x).-\frac12\log_3 x=\log_3\left(\frac{1}{\sqrt{x}}\right).$

Expanding a logarithm

To expand a logarithm means to rewrite it as a sum or difference of simpler logarithms. Work through multiplication and division first, then use the power law.

Example

Expanding a logarithmic expression

Expand

log⁡a(x3yz2).\log_a\left(\frac{x^3\sqrt{y}}{z^2}\right).

Use the quotient law to separate the numerator and denominator:
$log⁡a(x3y)−log⁡a(z2).\log_a\left(x^3\sqrt{y}\right)-\log_a(z^2).$
Use the product law on the numerator:
$log⁡a(x3)+log⁡a(y)−log⁡a(z2).\log_a(x^3)+\log_a(\sqrt{y})-\log_a(z^2).$
Write $y=y12\sqrt{y}=y^{\frac12}$ and apply the power law to each term:
$3log⁡ax+12log⁡ay−2log⁡az.3\log_a x+\frac12\log_a y-2\log_a z.$

Combining several logarithms

To combine logarithms, reverse the power law first so that coefficients become powers. Then use the product and quotient laws.

Example

Writing several terms as one logarithm

Write $2log⁡ax+12log⁡ay−log⁡az2\log_a x+\frac12\log_a y-\log_a z$ as a single logarithm.

Move each coefficient into its logarithm:
$log⁡a(x2)+log⁡a(y12)−log⁡az.\log_a(x^2)+\log_a(y^{\frac12})-\log_a z.$
Combine the added logarithms using the product law:
$log⁡a(x2y12)−log⁡az.\log_a(x^2y^{\frac12})-\log_a z.$
Apply the quotient law and replace $y12y^{\frac12}$ by $y\sqrt y$ :
$log⁡a(x2yz).\log_a\left(\frac{x^2\sqrt y}{z}\right).$

Common Mistake

Check the base and the domain

You may combine logarithms directly only when they have the same base. Every original logarithmic argument must also be positive, even if the final combined argument appears positive.

Tip

A useful order for combining

When combining logarithms, use powers first, then combine additions as products, and finally combine subtractions as quotients.

Exam technique

In the exam

Check that all logarithms have the same base before applying a law.
Move coefficients into logarithms as powers before trying to combine terms.
Preserve the order in subtraction: the subtracted logarithm gives the denominator.
When expanding, apply product and quotient laws before bringing powers down.
Check that every logarithmic argument satisfies the required positivity condition.

Self review

Check yourself

How would you write $3\log_a x-\log_a y$ as one logarithm?
What does $−12log⁡ax-\frac12\log_a x$ become when written as a single logarithm?
Why is $log_a x+\log_a y$ not equal to $log_a(x+y)$ ?

What you'll learn

What logarithms mean and when they are defined.
How to use the product, quotient and power laws.
How to expand one logarithm or combine several logarithms.
How the power law works with negative and fractional powers.

Before you start: what is a logarithm?

A logarithm is the inverse operation of raising a number to a power. It tells you which exponent produces a given number.

For example, because $2^3=8$ , it follows that $log_2 8=3$ .

Definition

Meaning of a logarithm

For a valid base $a$ ,

log⁡ax=b⟺ab=x.\log_a x=b \quad \Longleftrightarrow \quad a^b=x.

Here, $a$ is the base, $x$ is the argument and $b$ is the logarithm.

The base and argument must satisfy:

a>0,a≠1,x>0.a>0,\qquad a\neq 1,\qquad x>0.

In particular, you cannot take the logarithm of zero or a negative real number.

Example

Evaluating logarithms from powers

Evaluate $log_5 125$ and $log⁡4(12)\log_4 \left(\frac{1}{2}\right)$ .

Write 125 as a power of 5: $125=5^3$ . Therefore, $log_5 125=3$ .
Write $12\frac{1}{2}$ as a power of 4. Since $4−12=14=124^{-\frac12}=\frac{1}{\sqrt4}=\frac12$ , the required exponent is $−12-\frac12$ .
Therefore, $log⁡4(12)=−12\log_4 \left(\frac12\right)=-\frac12$ .

Why the laws work

The laws of logarithms follow from the laws of indices. For example, multiplying two powers with the same base means adding their exponents:

am⋅an=am+n.a^m\cdot a^n=a^{m+n}.

Logarithms reverse exponentiation, so multiplication inside a logarithm corresponds to addition outside it. Similarly, division corresponds to subtraction, and a power corresponds to multiplication.

Key Idea

Logarithms turn operations into simpler operations

For logarithms with the same base:

multiplication becomes addition;
division becomes subtraction;
a power becomes a multiplier.

The product law

The product law is

log⁡ax+log⁡ay≡log⁡a(xy).\log_a x+\log_a y\equiv\log_a(xy).

The symbol $≡\equiv$ means that the two expressions are identically equal whenever both sides are defined. The law works because if $x=a^m$ and $y=a^n$ , then

xy=a^m a^n=a^{m+n}.

Therefore, $log_a(xy)=m+n=\log_a x+\log_a y$ .

Example

Combining logarithms using the product law

Simplify $log_3 4+\log_3 9$ .

The logarithms have the same base and are being added, so apply the product law:
$log⁡34+log⁡39=log⁡3(4⋅9).\log_3 4+\log_3 9=\log_3(4\cdot 9).$
Calculate the product:
$log⁡3(4⋅9)=log⁡336.\log_3(4\cdot 9)=\log_3 36.$
Therefore, the combined expression is $log_3 36$ . It does not simplify to an integer because 36 is not an integer power of 3.

Common Mistake

Adding the arguments

The product law does not say $log_a x+\log_a y=\log_a(x+y)$ . Addition of logarithms produces a product of their arguments.

The quotient law

The quotient law is

log⁡ax−log⁡ay≡log⁡a(xy).\log_a x-\log_a y\equiv\log_a\left(\frac{x}{y}\right).

It comes from the index law

aman=am−n.\frac{a^m}{a^n}=a^{m-n}.

The order matters: the argument of the first logarithm becomes the numerator, and the argument of the subtracted logarithm becomes the denominator.

Example

Combining logarithms using the quotient law

Simplify $log_2 48-\log_2 3$ .

The logarithms have the same base and are being subtracted, so use the quotient law:
$log⁡248−log⁡23=log⁡2(483).\log_2 48-\log_2 3=\log_2\left(\frac{48}{3}\right).$
Simplify the quotient:
$log⁡2(483)=log⁡216.\log_2\left(\frac{48}{3}\right)=\log_2 16.$
Since $16=2^4$ , the expression equals 4.

Common Mistake

Reversing the quotient

In $log_a x-\log_a y$ , keep the same order: the result is $log⁡a(xy)\log_a\left(\frac{x}{y}\right)$ , not $log⁡a(yx)\log_a\left(\frac{y}{x}\right)$ .

The power law

The power law is

klog⁡ax≡log⁡a(xk).k\log_a x\equiv\log_a\left(x^k\right).

It allows a multiplier in front of a logarithm to become a power of its argument. The law can also be used in reverse:

log⁡a(xk)=klog⁡ax.\log_a\left(x^k\right)=k\log_a x.

This is especially useful when expanding logarithmic expressions.

Example

Using the power law in both directions

Rewrite $3\log_2 x$ as a single logarithm, and expand $log_5(y^7)$ .

Move the multiplier 3 into the first logarithm as a power:
$3\log_2 x=\log_2(x^3).$
Move the power 7 in the second expression outside as a multiplier:
$log_5(y^7)=7\log_5 y.$
These statements require the original logarithmic arguments to be positive, so $x > 0$ and $y^7>0$ .

Common Mistake

Multiplying the argument

The multiplier becomes an exponent, not a factor. Thus $k\log_a x=\log_a(x^k)$ , not $log_a(kx)$ .

Negative powers

The power law also applies when the multiplier is negative. In particular,

−log⁡ax=log⁡a(x−1)=log⁡a(1x).-\log_a x=\log_a\left(x^{-1}\right) =\log_a\left(\frac{1}{x}\right).

This is the case $k = - 1$ . It explains why changing the sign of a logarithm takes the reciprocal of its argument.

Example

Combining a logarithm with a negative coefficient

Write $log_a p-\log_a q$ as a single logarithm using the power and product laws.

Treat the subtraction as multiplication by negative one:
$log_a q=\log_a(q^{-1}).$
Use the product law to combine the logarithms:
$log_a p+\log_a(q^{-1})=\log_a(pq^{-1}).$
Since $q−1=1qq^{-1}=\frac1q$ , this becomes
$log⁡a(pq),\log_a\left(\frac{p}{q}\right),$
which agrees with the quotient law.

Fractional powers

A fractional power represents a root. For example,

x12=xandx−12=1x.x^{\frac12}=\sqrt{x} \qquad\text{and}\qquad x^{-\frac12}=\frac{1}{\sqrt{x}}.

Therefore, the power law gives

12log⁡ax=log⁡ax\frac12\log_a x=\log_a\sqrt{x}

and

−12log⁡ax=log⁡a(x−12)=log⁡a(1x).-\frac12\log_a x =\log_a\left(x^{-\frac12}\right) =\log_a\left(\frac{1}{\sqrt{x}}\right).

Example

Using a negative fractional coefficient

Rewrite $−12log⁡3x-\frac12\log_3 x$ as a single logarithm.

Use the power law to move the coefficient into the logarithm as an exponent:
$−12log⁡3x=log⁡3(x−12).-\frac12\log_3 x=\log_3\left(x^{-\frac12}\right).$
Interpret the exponent using the laws of indices:
$x−12=1x12=1x.x^{-\frac12}=\frac{1}{x^{\frac12}}=\frac{1}{\sqrt{x}}.$
Hence,
$−12log⁡3x=log⁡3(1x).-\frac12\log_3 x=\log_3\left(\frac{1}{\sqrt{x}}\right).$

Expanding a logarithm

To expand a logarithm means to rewrite it as a sum or difference of simpler logarithms. Work through multiplication and division first, then use the power law.

Example

Expanding a logarithmic expression

Expand

log⁡a(x3yz2).\log_a\left(\frac{x^3\sqrt{y}}{z^2}\right).

Use the quotient law to separate the numerator and denominator:
$log⁡a(x3y)−log⁡a(z2).\log_a\left(x^3\sqrt{y}\right)-\log_a(z^2).$
Use the product law on the numerator:
$log⁡a(x3)+log⁡a(y)−log⁡a(z2).\log_a(x^3)+\log_a(\sqrt{y})-\log_a(z^2).$
Write $y=y12\sqrt{y}=y^{\frac12}$ and apply the power law to each term:
$3log⁡ax+12log⁡ay−2log⁡az.3\log_a x+\frac12\log_a y-2\log_a z.$

Combining several logarithms

To combine logarithms, reverse the power law first so that coefficients become powers. Then use the product and quotient laws.

Example

Writing several terms as one logarithm

Write $2log⁡ax+12log⁡ay−log⁡az2\log_a x+\frac12\log_a y-\log_a z$ as a single logarithm.

Move each coefficient into its logarithm:
$log⁡a(x2)+log⁡a(y12)−log⁡az.\log_a(x^2)+\log_a(y^{\frac12})-\log_a z.$
Combine the added logarithms using the product law:
$log⁡a(x2y12)−log⁡az.\log_a(x^2y^{\frac12})-\log_a z.$
Apply the quotient law and replace $y12y^{\frac12}$ by $y\sqrt y$ :
$log⁡a(x2yz).\log_a\left(\frac{x^2\sqrt y}{z}\right).$

Common Mistake

Check the base and the domain

You may combine logarithms directly only when they have the same base. Every original logarithmic argument must also be positive, even if the final combined argument appears positive.

Tip

A useful order for combining

When combining logarithms, use powers first, then combine additions as products, and finally combine subtractions as quotients.

Exam technique

In the exam

Check that all logarithms have the same base before applying a law.
Move coefficients into logarithms as powers before trying to combine terms.
Preserve the order in subtraction: the subtracted logarithm gives the denominator.
When expanding, apply product and quotient laws before bringing powers down.
Check that every logarithmic argument satisfies the required positivity condition.

Self review

Check yourself

How would you write $3\log_a x-\log_a y$ as one logarithm?
What does $−12log⁡ax-\frac12\log_a x$ become when written as a single logarithm?
Why is $log_a x+\log_a y$ not equal to $log_a(x+y)$ ?

What you'll learn

Before you start: what is a logarithm?

Why the laws work

The product law

The quotient law

The power law

Negative powers

Fractional powers

Expanding a logarithm

Combining several logarithms

Review 1.9.4 Laws of logarithms by teaching Genie

1.9.4 Laws of logarithms

What you'll learn

Before you start: what is a logarithm?

Why the laws work

The product law

The quotient law

The power law

Negative powers

Fractional powers

Expanding a logarithm

Combining several logarithms

Review 1.9.4 Laws of logarithms by teaching Genie

1.9.4 Laws of logarithms

What you'll learn

Before you start: what is a logarithm?

Why the laws work

The product law

The quotient law

The power law

Negative powers

Fractional powers

Expanding a logarithm

Combining several logarithms

Review 1.9.4 Laws of logarithms by teaching Genie

1.9.4 Laws of logarithms Revision Guide

1.9.4 Laws of logarithms

What you'll learn

Before you start: what is a logarithm?

Why the laws work

The product law

The quotient law

The power law

Negative powers

Fractional powers

Expanding a logarithm

Combining several logarithms

Review 1.9.4 Laws of logarithms by teaching Genie

1.9.4 Laws of logarithms Revision Guide