Logarithm Rules (Properties)

Logarithm rules are the properties or the identities of the logarithm that are used to simplify complex logarithmic expressions and solve logarithmic equations involving variables. They are derived from the exponent rules, as they are just the opposite of writing an exponent.

Here is the list of all the logarithmic identities.

These rules are used to simplify complex logarithmic expressions and solve logarithmic equations involving variables.

Product Rule

It states that the logarithm of a product is the sum of the logarithms of the numbers being multiplied.

log_b(xy) = log_b(x) + log_b(y)

For example, log₈(63) = log₈(7 × 9) = log₈(7) + log₈(9)

Now, let us derive this rule.

Derivation

Let us assume log_b(x) = p and log_b(y) = q.

Converting the logarithms into exponential forms, we get

b^p = x …..(i)

b^q = y …..(ii)

Now, by multiplying (i) and (ii), we get

b^p × b^q = xy

⇒ b^{p + q} = xy …..(iii)

Converting (iii) into the logarithmic form, we get

log_b(xy) = p + q

By putting the values of p and q, we get

log_b(xy) = log_b(x) + log_b(y)

Quotient Rule

It states that the logarithm of the quotient is equal to the difference between the logarithm of the dividend minus the logarithm of the divisor.

${\log _{b}\left( \dfrac{x}{y}\right) =\log _{b}x-\log _{b}y}$

For example, ${\log _{6}\left( \dfrac{28}{25}\right) =\log _{6}\left( 28\right) -\log _{6}\left( 25\right)}$

Derivation

Let us assume log_b(x) = p and log_b(y) = q.

Converting the logarithms into exponential forms, we get

b^p = x …..(i) and b^q = y …..(ii)

On dividing (i) by (ii), we get

${\dfrac{b^{p}}{b^{q}}=\dfrac{x}{y}}$

⇒ ${b^{p-q}=\dfrac{x}{y}}$ …..(iii)

Converting (iii) into the logarithmic form, we get

${\log _{b}\left( \dfrac{x}{y}\right) =p-q}$

By putting the values of p and q, we get

${\log _{b}\left( \dfrac{x}{y}\right) =\log _{b}x-\log _{b}y}$

Power Rule

It states that the logarithm of a number raised to a power is the power times the logarithm of the number.

log_b(xⁿ) = n log_bx

For example, log₇(3⁸) = 8 log₇(3)

Derivation

Let us assume log_b(x) = p

Converting the logarithm into its exponential form, we get

b^p = x

Raising the power by n, we get

(b^p)ⁿ = xⁿ

⇒ b^np = xⁿ

Now, converting into the logarithmic form, we get

log_b(xⁿ) = np

By putting the value of p, we get

log_b(xⁿ) = n log_b(x)

Change of Base Rule

It states that the logarithm of a given number ‘x’ to a base ‘b’ can be expressed as the ratio of the logarithm of ‘x’ to any other base ‘c’ divided by the logarithm of ‘b’ to the base ‘c.’

${\log _{b}x=\dfrac{\log _{c}x}{\log _{c}b}}$

⇒ log_bx ⋅ log_cb = log_cx

For example, log₄(15) can be written as ${\dfrac{\log _{c}15}{\log _{c}4}}$, where ‘c’ is any other base.

Derivation

Let us assume log_b(x) = p, log_c(x) = r, and log_c(b) = s

Converting the logarithms into their exponential forms, we get

b^p = x …..(i)

c^r = x …..(ii)

c^s = b …..(iii)

From (i) and (ii), b^p = c^r …..(iv)

Substituting the value of (iii) in (iv), we get

(c^s)^p = c^r 

⇒ c^sp = c^r

⇒ sp = r

⇒ ${p=\dfrac{r}{s}}$

Now, substituting the values of p, r, and s, we get

${\log _{b}x=\dfrac{\log _{c}x}{\log _{c}b}}$

⇒ log_bx ⋅ log_cb = log_cx, another form of the rule.

Inverse Rule

It states that:

1. The logarithm of a base raised to a number is that number.

${\log _{b}\left( b^{x}\right) =x}$

2. A base raised to the logarithm of a number is equal to that number.

${b^{\log _{b}x}=x}$

For example, ${8^{\log _{8}5}=5}$ and ${\log _{3}\left( 3^{7}\right) =7}$

Derivation

Let us assume log_b(x) = p …..(i)

Converting the logarithm into its exponential form, we get

b^p = x …..(ii)

Now, by putting the value of p in (ii), we get

${b^{\log _{b}x}=x}$

Again, by putting the value of (ii) in (i), we get

log_b(b^p) = p

Thus, mathematically, it is ${\log _{b}\left( b^{x}\right) =x}$

Zero Rule

The logarithm of 1 to any base is zero.

log_b(1) = 0

For example, log₁₅(1) = 0, log₂(1) = 0

Derivation

Let us consider the logarithmic equation log_b(1) = x

Converting into its exponential form, we get

b^x = 1

⇒ b^x = b⁰

⇒ x = 0

Thus, log_b(1) = 0

Identity Rule

It states that the logarithm of a base to itself is 1.

log_b(b) = 1

For example, log₉(9) = 1, log₄(4) = 1

Derivation

Let us consider the logarithmic equation log_b(b) = x

Converting into its exponential form, we get

b^x = b

⇒ b^x = b¹

⇒ x = 1

Thus, log_b(b) = 1

Reciprocal Rule

It states that the logarithm of a number’s reciprocal is equal to the negative of the logarithm of that number. ${\log _{b}\left( \dfrac{1}{x}\right) =-\log _{b}\left( x\right)}$

For example, ${\log _{3}\left( \dfrac{1}{11}\right) =-\log _{3}\left( 11\right)}$

Derivation

Let ${x}$ and ${\dfrac{1}{x}}$ be the number and its reciprocal.

Since ${x\cdot \dfrac{1}{x}=1}$, then

${\log _{b}\left( x\cdot \dfrac{1}{x}\right) =\log _{b}\left( 1\right)}$

⇒ ${\log _{b}\left( x\right) +\log _{b}\left( \dfrac{1}{x}\right) =0}$

⇒ ${\log _{b}\left( \dfrac{1}{x}\right) =-\log _{b}\left( x\right)}$

Properties of Natural Logarithm

It states that the natural logarithm (denoted by log_e(x) or ln(x)) follows all the above properties of base logarithms. 

Here are some more special properties of natural logarithm:

Derivative Rule

It states that the derivative of a natural logarithm of a number with respect to it is equal to ${\dfrac{1}{x}}$.

${\dfrac{d}{dx}\left( \ln \left( x\right) \right) =\dfrac{1}{x}}$

Integration Rule

It states that the integration of ${\dfrac{1}{x}}$ with respect to ‘x’ is equal to the natural logarithm of the absolute value of x, plus a constant of integration ‘C.’

${\int \dfrac{1}{x}dx=\ln \left| x\right| +C}$, here ‘C’ is a constant.

Also, while integrating ln(x) with respect to x, the formula is as follows:

${\int \ln \left( x\right) dx=x\cdot \ln \left( x\right) -x+C}$

⇒ ${\int \ln \left( x\right) dx=x\left( \ln \left( x\right) -1\right) +C}$

Solved Example