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 addition of separate logs, while the quotient rule transforms division into subtraction. The power rule enables exponents to be moved in front of logs as multiplication. For complex bases, changing the base using the formula $log\left(m\right)/log\left(b\right)$ facilitates quick evaluations using calculators. Mastering these concepts enhances problem-solving skills in logarithmic functions.

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 of the fundamental properties is the product rule. This states that when you have a logarithm of a product, such as log_b(xy), it can be expanded into the sum of two logarithms: log_b(x) + log_b(y). For example, log₂(3x) can be expressed as log₂(3) + log₂(x).

Next is the quotient rule, which applies when dealing with the logarithm of a quotient. If you have log_b(x/y), this can be rewritten as log_b(x) - log_b(y). 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 simplified to 2 * ln(7).

By applying these rules—product, quotient, and power—you can effectively expand logarithmic expressions, making complex problems more manageable. Understanding these properties not only aids in solving logarithmic equations but also reinforces the connection between logarithms and exponents.

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 expand this expression into:

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), we can rewrite it as 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 careful attention to the base of the logs and the operations involved. For instance, when condensing 2 * ln(x) - ln(x + 2), we first apply the power rule to the first term, transforming it into ln(x²). The expression now reads:

ln(x²) - ln(x + 2).

Next, we apply the quotient rule, which states that the logarithm of a quotient can be expressed as the difference of the logarithms. This allows us to condense the expression into:

ln( (x²) / (x + 2) ).

By mastering these rules, you can confidently expand and condense any logarithmic expression you encounter. Practice with various examples will further solidify your understanding and application of these logarithmic properties.

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 non-standard 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 essential for this purpose and can be 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 a complex number, making direct calculation impractical.

For example, if you want 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 use the logarithm buttons available on most calculators to find the values easily.

Similarly, if you need to evaluate \(\log_{\pi}(9)\), you can convert it to base 10 as follows:

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

After entering this into a calculator, you would find that \(\log_{\pi}(9) \approx 1.92\). If you were to use natural logarithms instead, the expression would be:

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

Again, this would yield the same result of approximately 1.92, demonstrating that the choice of base does not affect the final outcome.

In another example, evaluating \(\log_{\sqrt{3}}(e)\) using natural logarithms gives:

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

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

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

Calculating this will provide a specific numerical value, further illustrating the versatility of the change of base formula.

Understanding how to manipulate logarithmic expressions using the change of base formula is a powerful tool in mathematics, allowing for quick evaluations and deeper insights into logarithmic relationships.

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:

How do you expand log base 2 of 3xy² using logarithmic properties?

To expand log base 2 of 3xy², we use the product and power rules of logarithms. First, recognize that 3xy² is a product of three terms: 3, x, and y². Using the product rule, we can write:

${\log }_2\left(3x{y}^2\right)={\log }_2\left(3\right)+{\log }_2\left(x\right)+{\log }_2\left(y^2\right)$

Next, apply the power rule to the term log base 2 of y²:

${\log }_2\left(y^2\right)=2{\log }_2\left(y\right)$

So, the fully expanded form is:

${\log }_2\left(3x{y}^2\right)={\log }_2\left(3\right)+{\log }_2\left(x\right)+2{\log }_2\left(y\right)$

How do you condense 2ln(x) - ln(x + 2) into a single logarithm?

To condense 2ln(x) - ln(x + 2) into a single logarithm, we use the power and quotient rules of logarithms. First, apply the power rule to 2ln(x):

$2\ln \left(x\right)=\ln \left(x^2\right)$

Next, use the quotient rule to combine the terms:

$\ln \left(x^2\right)-\ln (x+2)=\ln \left(\frac{x^2}{x+2}\right)$

So, the condensed form is:

$\ln \left(\frac{x^2}{x+2}\right)$

How do you change the base of log base 5 of 2 to base 10?

To change the base of log base 5 of 2 to base 10, we use the change of base formula:

${\log }_5\left(2\right)=\frac{{\log }_{10}\left(2\right)}{{\log }_{10}\left(5\right)}$

Since log base 10 is the common log, we can simplify this to:

$\log \left(2\right)/\log \left(5\right)$

Now, you can easily evaluate this using a calculator:

$\log \left(2\right)/\log \left(5\right)\approx 0.4307$

What is the power rule for logarithms and how is it used?

The power rule for logarithms states that if you have a logarithm of a number raised to a power, you can move the exponent to the front of the logarithm as a multiplier. Mathematically, it is expressed as:

${\log }_b\left(m^n\right)=n{\log }_b\left(m\right)$

For example, if you have log base 2 of 8³, you can use the power rule to simplify it:

${\log }_2\left(8^3\right)=3{\log }_2\left(8\right)$

This makes it easier to evaluate or further manipulate the logarithmic expression.

How do you evaluate log base 7 of 31 using a calculator?

To evaluate log base 7 of 31 using a calculator, you can use the change of base formula. Convert log base 7 of 31 to a base that your calculator can handle, such as base 10 or base e (natural log). The change of base formula is:

${\log }_7\left(31\right)=\frac{{\log }_{10}\left(31\right)}{{\log }_{10}\left(7\right)}$

Using common logs, this becomes:

$\log \left(31\right)/\log \left(7\right)$

Enter this into your calculator:

$\log \left(31\right)/\log \left(7\right)\approx 1.5132$