Properties of Logarithms: Videos & Practice Problems

Understanding logarithmic properties is essential for expanding and condensing log expressions. The product rule allows multiplication within logs to be expressed as the sum of separate logs, while the quotient rule transforms division into subtraction. The power rule enables exponents to be moved in front of logs as coefficients. For complex bases, changing the base using the formula $log\left(m\right)/log\left(b\right)$ simplifies calculations, facilitating quick evaluations with calculators.

Product, Quotient, and Power Rules of Logs

When expanding logarithmic expressions, it's essential to utilize the properties of logarithms, which are closely related to the properties of exponents. Understanding these relationships allows for the effective manipulation of log expressions without the need to memorize numerous new rules.

One fundamental property is the product rule. This states that when you have a logarithm of a product, such as log_b(m × n), you can separate it into the sum of two logarithms: log_b(m) + log_b(n). For example, log₂(3x) can be expanded to log₂(3) + log₂(x).

Next is the quotient rule, which applies when dealing with division in logarithms. If you have log_b(m/n), this can be expressed as the difference of two logarithms: log_b(m) - log_b(n). For instance, log₅(5/y) becomes log₅(5) - log₅(y).

Lastly, the power rule states that when you have a logarithm of a power, such as log_b(mⁿ), you can bring the exponent in front of the logarithm: n × log_b(m). For example, ln(7²) can be rewritten as 2 × ln(7).

By applying these rules—product, quotient, and power—you can effectively expand logarithmic expressions, simplifying complex problems into manageable parts. This understanding not only aids in solving logarithmic equations but also reinforces the connection between logarithmic and exponential functions.

Expand & Condense Log Expressions

Understanding logarithmic expressions involves mastering the rules for both expanding and condensing logs. When expanding a logarithmic expression, such as log₂(3xy²), we utilize the product rule, which states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors. Thus, we can rewrite the expression as:

log₂(3) + log₂(x) + log₂(y²).

Next, we can apply the power rule, which allows us to move the exponent in front of the logarithm. For the term log₂(y²), this becomes:

2 * log₂(y).

Therefore, the fully expanded form of the original expression is:

log₂(3) + log₂(x) + 2 * log₂(y).

On the other hand, condensing logarithmic expressions requires combining multiple logs into a single log. For instance, consider the expression 2 * ln(x) - ln(x + 2). First, we apply the power rule to the first term, transforming it into:

ln(x²).

Next, since we are subtracting the second logarithm, we apply the quotient rule, which states that the difference of logs corresponds to the logarithm of a quotient. Thus, we can condense the expression to:

ln(x² / (x + 2)).

In summary, the ability to expand and condense logarithmic expressions hinges on the application of the product rule, power rule, and quotient rule. Mastery of these rules enables you to manipulate logarithmic expressions effectively, whether you are breaking them down into simpler components or combining them into a single expression.

Write the log expression as a single log.
$\log_2\frac{1}{9x}+2\log_23x$

• A. $\log_2x$
• B. $\log_2\frac{1}{3x}$
• C. $\log_21$
• D. $\log_23x$

Answer: (A)

Write the log expression as a single log.
$\ln\frac{3x}{y}+2\ln2y-\ln4x$

• A. $\ln\frac{3xy}{4}$
• B. $\ln\left(12x^2\right)$
• C. $\ln\left(\frac32\right)$
• D. $\ln\left(3y\right)$

Answer: (D)

Write the single logarithm as a sum or difference of logs.
$\log_3\left(\frac{\sqrt{x}}{9y^2}\right)$

• A. $2\log_3x-2-\log_39y$
• B. $\frac12\log_3x-2-2\log_3y$
• C. $\frac12\log_3x+2\log_33y$
• D. $\frac12\log_3x-2\log_39y$

Answer: (B)

Write the single logarithm as a sum or difference of logs.
$\log_5\left(\frac{5\left(2x+3\right)^2}{x^3}\right)$

• A. $5+2\log_5\left(2x+3\right)-\log_53x$
• B. $2\log_5\left(2x+3\right)-3\log_5x$
• C. $1+2\log_5\left(2x+3\right)-3\log_5x$
• D. $\log_5\left(2x+3\right)-\log_5x$

Answer: (C)

Change of Base Property

When evaluating logarithms, especially those with unconventional bases like π, it can be beneficial to change the base to a more manageable one, such as 10 or e. This process allows for easier calculations using a calculator. The change of base formula is a key tool in this process, expressed as:

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

In this formula, \(b\) is the original base, \(m\) is the argument of the logarithm, and \(a\) is the new base you wish to use. The base \(b\) is placed in the denominator, while the argument \(m\) is in the numerator. This method is particularly useful when the base is not a standard value that can be directly input into a calculator.

For example, to evaluate \(\log_5(2)\) using base 10, you would rewrite it as:

$$\log_5(2) = \frac{\log(2)}{\log(5)}$$

Here, \(\log\) refers to the common logarithm (base 10). This allows you to easily compute the value using a calculator. Similarly, if you want to use natural logarithms (base e), the expression would be:

$$\log_5(2) = \frac{\ln(2)}{\ln(5)}$$

Both methods will yield the same result, demonstrating the flexibility of logarithmic calculations.

Consider another example: to evaluate \(\log_{\pi}(9)\), you can convert it to base 10:

$$\log_{\pi}(9) = \frac{\log(9)}{\log(\pi)}$$

Alternatively, using natural logarithms, it can be expressed as:

$$\log_{\pi}(9) = \frac{\ln(9)}{\ln(\pi)}$$

Both approaches will provide the same numerical result, reinforcing the concept that the choice of base does not affect the outcome of the logarithmic evaluation.

In a more complex scenario, if you need to evaluate \(\log_{\sqrt{3}}(e)\) using natural logs, you can express it as:

$$\log_{\sqrt{3}}(e) = \frac{\ln(e)}{\ln(\sqrt{3})}$$

Since \(\ln(e) = 1\), this simplifies to:

$$\log_{\sqrt{3}}(e) = \frac{1}{\ln(\sqrt{3})}$$

By applying the change of base formula, you can efficiently evaluate logarithms with any base, ensuring accurate results with minimal effort. Practice with various examples will enhance your proficiency in using this technique.

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.
$\log_317$

• A. 0.39
• B. 2.58
• C. 1.23
• D. 0.48

Answer: (B)

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.
$\log_967$

• A. 1.91
• B. 0.52
• C. 0.95
• D. 1.83

Answer: (A)

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.
$\log_841$

• A. 1.61
• B. 0.9
• C. 0.56
• D. 1.79

Answer: (D)

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log. 
$\log_23789$

• A. 0.08
• B. 11.89
• C. 3.58
• D. 0.30

Answer: (B)

Here’s what students ask on this topic:

What are the properties of logarithms and how are they used to expand log expressions?

The properties of logarithms include the product rule, quotient rule, and power rule. The product rule states that log_b(mn) = log_b(m) + log_b(n), allowing multiplication within logs to be expressed as the sum of separate logs. The quotient rule states that log_b(m/n) = log_b(m) - log_b(n), transforming division into subtraction. The power rule states that log_b(mⁿ) = n * log_b(m), enabling exponents to be moved in front of logs as coefficients. These properties are used to expand log expressions by breaking down complex logs into simpler components.

How do you condense multiple logarithmic expressions into a single log?

To condense multiple logarithmic expressions into a single log, you use the properties of logarithms in reverse. First, ensure all logs have the same base. Apply the power rule to move coefficients back to exponents. Then, use the product rule to combine logs that are added: log_b(m) + log_b(n) = log_b(mn). Use the quotient rule to combine logs that are subtracted: log_b(m) - log_b(n) = log_b(m/n). This process simplifies multiple logs into a single log expression.

How do you change the base of a logarithm for easier calculation?

To change the base of a logarithm, use the change of base formula: log_b(m) = log_a(m) / log_a(b). This formula allows you to convert a log with an inconvenient base to a more manageable base, such as 10 or e, which are common on calculators. For example, log₅(2) can be changed to log(2) / log(5) using base 10. This makes it easier to evaluate using a calculator.

What is the product rule for logarithms and how is it applied?

The product rule for logarithms states that log_b(mn) = log_b(m) + log_b(n). This rule allows you to express the logarithm of a product as the sum of the logarithms of its factors. For example, log₂(3x) can be expanded to log₂(3) + log₂(x). This property is useful for simplifying and solving logarithmic expressions.

How do you use the power rule to simplify logarithmic expressions?

The power rule for logarithms states that log_b(mⁿ) = n * log_b(m). This rule allows you to move an exponent in the argument of a log to the front as a coefficient. For example, log₂(x³) can be simplified to 3 * log₂(x). This property is particularly useful for simplifying expressions with exponents.