10.4 Use the Properties of Logarithms

Properties of Logarithms

Logarithm properties let you break apart complicated expressions into simpler pieces. Instead of working with the log of a big product, quotient, or power, you can rewrite it as addition, subtraction, or multiplication. These properties show up constantly when you're solving exponential and logarithmic equations, so getting comfortable with them now will pay off throughout the rest of the course.

Properties of Logarithms

The three core properties

Product Property:

$\log_b(M \cdot N) = \log_b(M) + \log_b(N)$

The log of a product equals the sum of the logs of each factor. This works because logarithms are really about exponents, and when you multiply two exponential expressions with the same base, you add the exponents.

• Example: $\log_3(8x) = \log_3(8) + \log_3(x)$
• Example: $\log_5(2 \cdot 7) = \log_5(2) + \log_5(7)$

Quotient Property:

$\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

The log of a quotient equals the log of the numerator minus the log of the denominator. This mirrors how dividing exponential expressions means subtracting exponents.

• Example: $\log_3\left(\frac{27}{9}\right) = \log_3(27) - \log_3(9) = 3 - 2 = 1$
• Example: $\log_4\left(\frac{x}{5}\right) = \log_4(x) - \log_4(5)$

Power Property:

$\log_b(M^n) = n \cdot \log_b(M)$

The log of a number raised to a power equals the exponent times the log of the base number. This property is especially useful because it pulls exponents out front, turning them into simple multiplication.

• Example: $\log_2(x^5) = 5 \cdot \log_2(x)$
• Example: $\log(10^3) = 3 \cdot \log(10) = 3 \cdot 1 = 3$

Change-of-base formula

Most calculators only have buttons for $\log$ (base 10) and $\ln$ (base $e$). The change-of-base formula lets you evaluate a logarithm with any base by converting it:

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

You can choose any new base $a$, but base 10 or base $e$ are the practical choices since those are on your calculator.

• Example: To evaluate $\log_5(20)$, rewrite it as $\frac{\log(20)}{\log(5)} = \frac{1.3010}{0.6990} \approx 1.861$
• You could also use natural log: $\frac{\ln(20)}{\ln(5)}$ gives the same result.

Simplification of complex logarithms

When you need to expand or simplify a logarithmic expression, apply the properties in a systematic order:

1. Use the product property to split logs of products into sums.
2. Use the quotient property to split logs of fractions into differences.
3. Use the power property to bring exponents down in front as coefficients.
4. Combine like terms if possible.

Example: Expand $\log_2\left(\frac{x^3 y}{z^2}\right)$

1. Apply the quotient property: $\log_2(x^3 y) - \log_2(z^2)$

2. Apply the product property to the first term: $\log_2(x^3) + \log_2(y) - \log_2(z^2)$

3. Apply the power property: $3\log_2(x) + \log_2(y) - 2\log_2(z)$

When different bases appear, use the change-of-base formula first to get everything into the same base, then apply the other properties.

• Example: If an expression contains both $\log_3(x^2)$ and $\log_9(z)$, convert $\log_9(z)$ to base 3 using the change-of-base formula: $\log_9(z) = \frac{\log_3(z)}{\log_3(9)} = \frac{\log_3(z)}{2}$. Now all terms share base 3 and you can combine them.

Logarithms and functions

Logarithmic functions are the inverses of exponential functions. If $f(x) = b^x$, then $f^{-1}(x) = \log_b(x)$. The base $b$ must be positive and not equal to 1.

• The domain of $\log_b(x)$ is all positive real numbers ($x > 0$). You cannot take the log of zero or a negative number.
• The range is all real numbers. A logarithm can output any value from $-\infty$ to $\infty$.

This inverse relationship is why logarithm properties mirror exponent rules: the product property corresponds to the rule $b^m \cdot b^n = b^{m+n}$, the quotient property corresponds to $\frac{b^m}{b^n} = b^{m-n}$, and the power property corresponds to $(b^m)^n = b^{mn}$.