Lesson 6.5 : Logarithmic Properties

New Concept

This lesson unveils key properties—the product, quotient, and power rules. You'll learn how to expand complex logs into simpler parts or condense multiple logs back into a single, powerful expression.

What’s next

Next, you’ll apply them through interactive examples and a series of practice cards, mastering expansion and condensation techniques.

Property

The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.

To use this rule, first factor the argument completely, expressing each whole number factor as a product of primes. Then, write the equivalent expression by summing the logarithms of each factor.

Examples

• Expand $\log_5 (15a(2a+3))$. This becomes $\log_5(3 \cdot 5 \cdot a \cdot (2a+3))$, which expands to $\log_5(3) + \log_5(5) + \log_5(a) + \log_5(2a+3)$.
• Expand $\log_2 (8x)$. This expands to $\log_2(8) + \log_2(x)$. Since $8 = 2^3$, this simplifies to $3 + \log_2(x)$.
• Expand $\ln(7xy)$. This is a product of three factors, so it expands to $\ln(7) + \ln(x) + \ln(y)$.

Explanation

Think of logs as exponents. When you multiply numbers with the same base, you add their exponents. The product rule for logs does the same thing: the logarithm of a product becomes the sum of the individual logarithms.

Property

The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.

To apply this, first express the argument in lowest terms by factoring and canceling. Then, write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.

Examples

• Expand $\log_4 \left( \frac{x}{64} \right)$. Using the quotient rule, this is $\log_4(x) - \log_4(64)$. Since $64 = 4^3$, this simplifies to $\log_4(x) - 3$.
• Expand $\ln \left( \frac{3x+6}{x+2} \right)$. First, factor the numerator to get $\ln \left( \frac{3(x+2)}{x+2} \right)$, which simplifies to $\ln(3)$ after canceling the common factor.
• Expand $\log \left( \frac{100}{y} \right)$. This becomes $\log(100) - \log(y)$, which simplifies to $2 - \log(y)$ because $\log(100) = \log(10^2) = 2$.

Explanation

Just like dividing powers means subtracting exponents, the logarithm of a quotient becomes a difference. It's simply the logarithm of the numerator minus the logarithm of the denominator.

Property

The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

To use this rule, express the argument as a power if needed. Then, write the equivalent expression by multiplying the exponent times the logarithm of the base.

Examples

• Expand $\log_3(x^7)$. The exponent 7 moves to the front, giving $7 \log_3(x)$.
• Expand $\log_5(125)$. First, rewrite 125 as $5^3$. The expression becomes $\log_5(5^3)$. Using the power rule gives $3 \log_5(5)$, which simplifies to $3 \cdot 1 = 3$.
• Rewrite $6 \ln(y)$ as a single logarithm using the power rule in reverse. The factor 6 becomes the exponent on the argument, resulting in $\ln(y^6)$.

Explanation

This rule lets you move an exponent from inside a logarithm to the front as a multiplier. It's a powerful shortcut for simplifying expressions involving powers and roots, since a root is just a fractional exponent.

Property

To expand logarithmic expressions, the product rule, quotient rule, and power rule are applied together to rewrite a single complex logarithm into a sum or difference of simpler logs.
It is important to remember that there is no way to expand the logarithm of a sum or difference, such as $\ln(x^2 + y^2)$.

Examples

• Rewrite $\ln \left( \frac{a^3 b}{5} \right)$ as a sum or difference of logs. This expands to $\ln(a^3) + \ln(b) - \ln(5)$, which simplifies to $3 \ln(a) + \ln(b) - \ln(5)$.
• Expand $\log(\sqrt{y})$. This is equivalent to $\log(y^{\frac{1}{2}})$, which simplifies to $\frac{1}{2}\log(y)$.
• Expand $\log_2 \left( \frac{8x^4(y-1)}{z+3} \right)$. This becomes $\log_2(8) + \log_2(x^4) + \log_2(y-1) - \log_2(z+3)$, simplifying to $3 + 4\log_2(x) + \log_2(y-1) - \log_2(z+3)$.

Explanation

Expanding is like taking apart a complex logarithmic expression. You apply the product, quotient, and power rules in combination to break one big logarithm into several smaller, easier-to-manage pieces.

Property

To condense a sum, difference, or product of logarithms with the same base into a single logarithm, use the rules in reverse order:

1. Apply the power property first. Rewrite terms like $n \log_b M$ as $\log_b (M^n)$.
2. Next, apply the product property. Rewrite sums of logs like $\log_b M + \log_b N$ as $\log_b (MN)$.
3. Apply the quotient property last. Rewrite differences of logs like $\log_b M - \log_b N$ as $\log_b (\frac{M}{N})$.

Examples

• Write $\log_4(6) + \log_4(5) - \log_4(3)$ as a single logarithm. This condenses to $\log_4(6 \cdot 5) - \log_4(3) = \log_4(30) - \log_4(3) = \log_4(\frac{30}{3}) = \log_4(10)$.
• Condense $3 \log(x) + \frac{1}{2} \log(y) - \log(z)$. Applying the power rule first gives $\log(x^3) + \log(y^{\frac{1}{2}}) - \log(z)$, which condenses to $\log \left( \frac{x^3 \sqrt{y}}{z} \right)$.
• Condense $2 \ln(x-1) - \ln(x^2-1)$. This becomes $\ln((x-1)^2) - \ln((x-1)(x+1))$, which simplifies to $\ln \left( \frac{(x-1)^2}{(x-1)(x+1)} \right) = \ln \left( \frac{x-1}{x+1} \right)$.

Explanation

Condensing is the opposite of expanding. You use the log rules to combine multiple logarithms back into one single, compact logarithm. This is a key step for solving many logarithmic equations.

Property

The change-of-base formula can be used to evaluate a logarithm with any base. For any positive real numbers $M, b,$ and $n$, where $n \neq 1$ and $b \neq 1$,

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

Examples

• Change $\log_7(12)$ to a quotient of natural logarithms. Using the formula, this becomes $\frac{\ln(12)}{\ln(7)}$.
• Evaluate $\log_4(20)$ using a calculator. Use the formula $\log_4(20) = \frac{\log(20)}{\log(4)}$. On a calculator, this is approximately $2.16096$.
• Change $\log_9(27)$ to a quotient of common logs. The expression becomes $\frac{\log(27)}{\log(9)}$, which evaluates to $1.5$ since $9^{1.5} = 27$.

Explanation

Your calculator likely only has log (base 10) and ln (base e) buttons. This formula is a bridge, letting you convert any logarithm into a fraction of logs that your calculator can actually compute.