Lesson 10.4: Use the Properties of Logarithms

New Concept

This lesson introduces the key properties of logarithms—Product, Quotient, and Power. You'll learn how to expand and condense logarithmic expressions, a crucial skill for solving complex logarithmic and exponential equations, and how to evaluate any logarithm.

What’s next

Now that you have the basics, you'll work through interactive examples of expanding and condensing logarithms, followed by practice problems to master these new properties.

Property

$\\log_a 1 = 0$
$\\log_a a = 1$

Examples

• Evaluate $\log_{12} 1$. Using the property $\log_a 1 = 0$, the result is $0$.
• Evaluate $\log_8 8$. Using the property $\log_a a = 1$, the result is $1$.
• Evaluate $\ln 1$. Since the base is $e$, this follows $\log_a 1 = 0$, so the result is $0$.

Explanation

These two properties are shortcuts derived from the definition of logarithms. A logarithm asks for the exponent. Since any base to the power of 0 is 1, $\log_a 1$ is always 0. Since any base to the power of 1 is itself, $\log_a a$ is always 1.

Property

For $a > 0, x > 0$ and $a \neq 1$,
$a^{\log_a x} = x$
$\\log_a a^x = x$

Examples

• Evaluate $6^{\log_6 11}$. Using the property $a^{\log_a x} = x$, the result is $11$.
• Evaluate $\log_4 4^9$. Using the property $\log_a a^x = x$, the result is $9$.
• Evaluate $e^{\ln 12}$. Since $\ln$ is $\log_e$, this simplifies to $12$.

Explanation

These properties show that exponentiation and logarithms are inverse operations. When the bases match, one function 'undoes' the other, leaving just the argument or the exponent. It's like adding and then subtracting the same number.

Property

If $M > 0$, $N > 0$, $a > 0$ and $a \neq 1$, then,

The logarithm of a product is the sum of the logarithms.

Examples

• Expand $\log_5(25x)$. This becomes $\log_5 25 + \log_5 x$, which simplifies to $2 + \log_5 x$.
• Expand $\log_b (3yz)$. This becomes $\log_b 3 + \log_b y + \log_b z$.
• Expand $\ln(ex)$. This becomes $\ln e + \ln x$, which simplifies to $1 + \ln x$.

Explanation

This property turns multiplication inside a logarithm into the addition of separate logarithms. It mirrors the exponent rule where multiplying powers with the same base means you add their exponents. Since logs are exponents, this rule applies.

Property

If $M > 0$, $N > 0$, $a > 0$ and $a \neq 1$, then,

The logarithm of a quotient is the difference of the logarithms.

Examples

• Expand $\log_4 \frac{x}{64}$. This becomes $\log_4 x - \log_4 64$, which simplifies to $\log_4 x - 3$.
• Expand $\log \frac{y}{1000}$. This becomes $\log y - \log 1000$, which simplifies to $\log y - 3$.
• Expand $\ln \frac{e^5}{2}$. This becomes $\ln e^5 - \ln 2$, which simplifies to $5 - \ln 2$.

Explanation

This property transforms division inside a logarithm into subtraction of two logarithms. It is the counterpart to the exponent rule for division, where you subtract exponents when dividing powers with the same base.

Property

If $M > 0$, $a > 0$, $a \neq 1$ and $p$ is any real number then,

The log of a number raised to a power is the product of the power times the log of the number.

Examples

• Expand $\log_3 9^5$. This becomes $5 \log_3 9$, which simplifies to $5 \cdot 2 = 10$.
• Expand $\ln \sqrt{x}$. This is $\ln x^{\frac{1}{2}}$, which becomes $\frac{1}{2} \ln x$.
• Expand $\log_b y^{-3}$. This becomes $-3 \log_b y$.

Explanation

This powerful rule allows you to move an exponent from inside a logarithm to the front as a coefficient. It's incredibly useful for simplifying expressions and solving equations where the variable is in the exponent.

Property

To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

Examples

• Condense $2\log_3 x + 5\log_3 y$. This becomes $\log_3 x^2 + \log_3 y^5 = \log_3(x^2 y^5)$.
• Condense $\ln 10 - 3\ln z$. This becomes $\ln 10 - \ln z^3 = \ln(\frac{10}{z^3})$.
• Condense $\log_2 40 - \log_2 5$. Using the quotient property, this becomes $\log_2(\frac{40}{5}) = \log_2 8 = 3$.

Explanation

Condensing is the reverse of expanding. You use the log properties backwards to combine a sum or difference of logs into a single, more compact logarithm. This is key for simplifying and solving log equations.

Property

For any logarithmic bases $a, b$ and $M > 0$,

For use with calculators, we use base 10 or base $e$:

Examples

• To approximate $\log_6 40$, use the formula $\frac{\log 40}{\log 6} \approx \frac{1.602}{0.778} \approx 2.059$.
• To approximate $\log_2 100$, use the formula $\frac{\ln 100}{\ln 2} \approx \frac{4.605}{0.693} \approx 6.645$.
• Rewrite $\log_k 9$ using base 10. The expression becomes $\frac{\log 9}{\log k}$.

Explanation

Since calculators typically only have 'log' (base 10) and 'ln' (base e) buttons, this formula is essential. It lets you convert a logarithm of any base into a fraction of logs that your calculator can handle.