MTH120 College Algebra Chapter 6.5

 6.5 Logarithmic Properties

Logarithmic properties are a set of rules and properties that help simplify and manipulate logarithmic expressions. Understanding these properties is essential for solving logarithmic equations and performing various calculations. Here are some important logarithmic properties:

1. Product Rule:

   • ${\log }_{�}(��)={\log }_{�}(�)+{\log }_{�}(�)$
   • The logarithm of the product of two numbers is equal to the sum of the logarithms of those numbers.

2. Quotient Rule:

   • ${\log }_{�}\left(\frac{�}{�}\right)={\log }_{�}(�)-{\log }_{�}(�)$
   • The logarithm of the quotient of two numbers is equal to the difference of the logarithms of those numbers.

3. Power Rule:

   • ${\log }_{�}({�}^{�})=�\cdot {\log }_{�}(�)$
   • The logarithm of a number raised to a power is equal to the power times the logarithm of the base.

4. Change of Base Formula:

   • ${\log }_{�}(�)=\frac{{\log }_{�}(�)}{{\log }_{�}(�)}$
   • Allows you to change the base of a logarithm to a different base.

5. Zero Exponent Rule:

   • ${\log }_{�}(1)=0$
   • The logarithm of 1 to any base is always 0.

6. Negative Exponent Rule:

   • ${\log }_{�}\left(\frac{1}{�}\right)=-{\log }_{�}(�)$
   • The logarithm of the reciprocal of a number is equal to the negative of the logarithm of the original number.

7. Equality Rule:

   • If ${\log }_{�}(�)={\log }_{�}(�)$, then $�=�$.

8. Sum and Difference Rule:

   • ${\log }_{�}(�+�)$ and ${\log }_{�}(�-�)$ cannot be simplified further unless additional information is provided.

These properties are especially useful when working with logarithmic equations and simplifying expressions. They allow you to condense or expand logarithmic expressions, making calculations more manageable.

The Product Rule for logarithms states that the logarithm of a product of two numbers is equal to the sum of the logarithms of those two numbers. In mathematical notation, it is expressed as:

${\log }_{�}(��)={\log }_{�}(�)+{\log }_{�}(�)$

Here are some examples of how to use the Product Rule for logarithms:

Example 1: Let's say we have the expression ${\log }_2(4\cdot 8)$. We can use the Product Rule to simplify this expression:

${\log }_2(4\cdot 8)={\log }_2(32)={\log }_2(2^5)=5\cdot {\log }_2(2)=5\cdot 1=5$

So, ${\log }_2(4\cdot 8)=5$.

Example 2: Suppose you have ${\log }_{10}(1000\cdot 0.1)$. Using the Product Rule:

${\log }_{10}(1000\cdot 0.1)={\log }_{10}(1000)+{\log }_{10}(0.1)$

Now, we know that ${\log }_{10}(1000)=3$ because $10^3=1000$, and ${\log }_{10}(0.1)=-1$ because $10^{-1}=0.1$.

So, ${\log }_{10}(1000\cdot 0.1)=3+(-1)=2$.

Example 3: Suppose you want to calculate ${\log }_3(27\cdot 9)$. Using the Product Rule:

${\log }_3(27\cdot 9)={\log }_3(27)+{\log }_3(9)$

We know that ${\log }_3(27)=3$ because $3^3=27$, and ${\log }_3(9)=2$ because $3^2=9$.

So, ${\log }_3(27\cdot 9)=3+2=5$.

These examples illustrate how to apply the Product Rule for logarithms to simplify expressions involving the product of numbers. By breaking down the products into separate logarithmic terms and summing them, you can make complex logarithmic expressions more manageable.

The Product Rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. In mathematical notation, it is expressed as:

${\log }_{�}(��)={\log }_{�}(�)+{\log }_{�}(�)$

So, if you have the logarithm of a product, you can use the Product Rule to write it as an equivalent sum of logarithms. Here are some examples:

Example 1: Given ${\log }_2(8\cdot 16)$, you can use the Product Rule to write it as a sum of logarithms:

${\log }_2(8\cdot 16)={\log }_2(8)+{\log }_2(16)$

Example 2: Given ${\log }_{10}(100\cdot 1000)$, you can use the Product Rule to write it as a sum of logarithms:

${\log }_{10}(100\cdot 1000)={\log }_{10}(100)+{\log }_{10}(1000)$

Example 3: Given ${\log }_3(9\cdot 27)$, you can use the Product Rule to write it as a sum of logarithms:

${\log }_3(9\cdot 27)={\log }_3(9)+{\log }_3(27)$

In each of these examples, we applied the Product Rule for logarithms to rewrite the logarithm of a product as the sum of logarithms of the individual factors. This can be particularly useful for simplifying logarithmic expressions and calculations.

The Quotient Rule for logarithms states that the logarithm of a quotient (division) of two numbers is equal to the difference of the logarithms of those two numbers. In mathematical notation, it is expressed as:

${\log }_{�}\left(\frac{�}{�}\right)={\log }_{�}(�)-{\log }_{�}(�)$

Here are some examples of how to use the Quotient Rule for logarithms:

Example 1: Suppose you have ${\log }_5\left(\frac{125}{25}\right)$. You can use the Quotient Rule to simplify it:

${\log }_5\left(\frac{125}{25}\right)={\log }_5(125)-{\log }_5(25)$

We know that ${\log }_5(125)=3$ because $5^3=125$, and ${\log }_5(25)=2$ because $5^2=25$.

So, ${\log }_5\left(\frac{125}{25}\right)=3-2=1$.

Example 2: Given ${\log }_{10}\left(\frac{1000}{10}\right)$, you can use the Quotient Rule to simplify it:

${\log }_{10}\left(\frac{1000}{10}\right)={\log }_{10}(1000)-{\log }_{10}(10)$

Now, ${\log }_{10}(1000)=3$ because $10^3=1000$, and ${\log }_{10}(10)=1$ because $10^1=10$.

So, ${\log }_{10}\left(\frac{1000}{10}\right)=3-1=2$.

Example 3: Suppose you want to calculate ${\log }_2\left(\frac{64}{8}\right)$. Using the Quotient Rule:

${\log }_2\left(\frac{64}{8}\right)={\log }_2(64)-{\log }_2(8)$

We know that ${\log }_2(64)=6$ because $2^6=64$, and ${\log }_2(8)=3$ because $2^3=8$.

So, ${\log }_2\left(\frac{64}{8}\right)=6-3=3$.

These examples illustrate how to apply the Quotient Rule for logarithms to simplify expressions involving the division of numbers. By breaking down the division into separate logarithmic terms and subtracting them, you can make complex logarithmic expressions more manageable.

The Power Rule for logarithms states that the logarithm of a number raised to a power is equal to the power times the logarithm of the base. In mathematical notation, it is expressed as:

${\log }_{�}({�}^{�})=�\cdot {\log }_{�}(�)$

Here are some examples of how to use the Power Rule for logarithms:

Example 1: Suppose you have ${\log }_2(8^3)$. You can use the Power Rule to simplify it:

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

Now, ${\log }_2(8)=3$ because $2^3=8$.

So, ${\log }_2(8^3)=3\cdot 3=9$.

Example 2: Given ${\log }_{10}(100^2)$, you can use the Power Rule to simplify it:

${\log }_{10}(100^2)=2\cdot {\log }_{10}(100)$

We know that ${\log }_{10}(100)=2$ because $10^2=100$.

So, ${\log }_{10}(100^2)=2\cdot 2=4$.

Example 3: Suppose you want to calculate ${\log }_3\left(5^4\right)$. Using the Power Rule:

${\log }_3\left(5^4\right)=4\cdot {\log }_3(5)$

In this case, you do not need to simplify further because you don't know the exact values. This is a valid simplification based on the Power Rule.

These examples illustrate how to apply the Power Rule for logarithms to simplify expressions involving the exponentiation of numbers. By bringing the exponent down as a coefficient of the logarithm, you can make logarithmic expressions more manageable.

Expanding logarithmic expressions involves using the properties of logarithms to break down a single logarithmic term into multiple terms. The key properties used for expansion are the Product Rule, Quotient Rule, and Power Rule for logarithms. Here's how to expand logarithmic expressions:

Product Rule:

• ${\log }_{�}(��)={\log }_{�}(�)+{\log }_{�}(�)$

Quotient Rule:

• ${\log }_{�}\left(\frac{�}{�}\right)={\log }_{�}(�)-{\log }_{�}(�)$

Power Rule:

• ${\log }_{�}({�}^{�})=�\cdot {\log }_{�}(�)$

Steps for Expanding Logarithmic Expressions:

1. Identify the expression you want to expand. It should be in the form of a single logarithmic term.

2. Apply the appropriate logarithmic property (Product Rule, Quotient Rule, or Power Rule) based on the structure of the expression.

3. Break down the expression into multiple terms, following the properties you applied.

4. Write the expanded expression as a sum or difference of logarithmic terms.

Example 1: Expand ${\log }_2(8\cdot 16)$.

Using the Product Rule: ${\log }_2(8\cdot 16)={\log }_2(8)+{\log }_2(16)$

Now, simplify each term: ${\log }_2(8)=3$ because $2^3=8$. ${\log }_2(16)=4$ because $2^4=16$.

So, the expanded expression is: ${\log }_2(8\cdot 16)=3+4=7$.

Example 2: Expand ${\log }_{10}\left(\frac{1000}{10}\right)$.

Using the Quotient Rule: ${\log }_{10}\left(\frac{1000}{10}\right)={\log }_{10}(1000)-{\log }_{10}(10)$

Now, simplify each term: ${\log }_{10}(1000)=3$ because $10^3=1000$. ${\log }_{10}(10)=1$ because $10^1=10$.

So, the expanded expression is: ${\log }_{10}\left(\frac{1000}{10}\right)=3-1=2$.

By applying the appropriate logarithmic properties, you can expand logarithmic expressions to make calculations or simplifications easier.

Expanding complex logarithmic expressions involves using the properties of logarithms to break down an expression with multiple terms and simplify it. Here are some examples of how to expand complex logarithmic expressions:

Example 1: Expand ${\log }_2(8\cdot 16\cdot 32)$.

Using the Product Rule repeatedly: ${\log }_2(8\cdot 16\cdot 32)={\log }_2(8)+{\log }_2(16)+{\log }_2(32)$

Now, simplify each term: ${\log }_2(8)=3$ because $2^3=8$. ${\log }_2(16)=4$ because $2^4=16$. ${\log }_2(32)=5$ because $2^5=32$.

So, the expanded expression is: ${\log }_2(8\cdot 16\cdot 32)=3+4+5=12$.

Example 2: Expand ${\log }_{10}\left(\frac{1000}{10\cdot 5}\right)$.

Using the Quotient Rule: ${\log }_{10}\left(\frac{1000}{10\cdot 5}\right)={\log }_{10}(1000)-{\log }_{10}(10)-{\log }_{10}(5)$

Now, simplify each term: ${\log }_{10}(1000)=3$ because $10^3=1000$. ${\log }_{10}(10)=1$ because $10^1=10$. ${\log }_{10}(5)={\log }_{10}(5)$ because it cannot be simplified further.

So, the expanded expression is: ${\log }_{10}\left(\frac{1000}{10\cdot 5}\right)=3-1-{\log }_{10}(5)$.

Example 3: Expand ${\log }_3(27^2\cdot 9^3\cdot 81)$.

Using the Power Rule and the Product Rule: ${\log }_3(27^2\cdot 9^3\cdot 81)=2\cdot {\log }_3(27)+3\cdot {\log }_3(9)+{\log }_3(81)$

Now, simplify each term: ${\log }_3(27)=3$ because $3^3=27$. ${\log }_3(9)=2$ because $3^2=9$. ${\log }_3(81)=4$ because $3^4=81$.

So, the expanded expression is: ${\log }_3(27^2\cdot 9^3\cdot 81)=2\cdot 3+3\cdot 2+4=12$.

By applying the appropriate logarithmic properties, you can expand complex logarithmic expressions and simplify them step by step. This can be particularly useful for solving logarithmic equations or simplifying mathematical expressions.

Condensing logarithmic expressions involves using the properties of logarithms to combine multiple terms into a single, simplified expression. The key properties used for condensation are the Product Rule, Quotient Rule, and Power Rule for logarithms. Here's how to condense logarithmic expressions:

Product Rule:

• ${\log }_{�}(��)={\log }_{�}(�)+{\log }_{�}(�)$

Quotient Rule:

• ${\log }_{�}\left(\frac{�}{�}\right)={\log }_{�}(�)-{\log }_{�}(�)$

Power Rule:

• ${\log }_{�}({�}^{�})=�\cdot {\log }_{�}(�)$

Steps for Condensing Logarithmic Expressions:

1. Identify the expression you want to condense. It should be in the form of a sum or difference of logarithmic terms.

2. Apply the appropriate logarithmic property (Product Rule, Quotient Rule, or Power Rule) to combine the terms into a single logarithmic term.

3. Write the condensed expression as a single logarithmic term.

Example 1: Condense ${\log }_2(8)+{\log }_2(16)+{\log }_2(32)$.

Using the Product Rule: ${\log }_2(8)+{\log }_2(16)+{\log }_2(32)={\log }_2(8\cdot 16\cdot 32)$

Now, simplify the product: ${\log }_2(8\cdot 16\cdot 32)={\log }_2(32768)$

So, the condensed expression is: ${\log }_2(8)+{\log }_2(16)+{\log }_2(32)={\log }_2(32768)$.

Example 2: Condense ${\log }_{10}(1000)-{\log }_{10}(10)-{\log }_{10}(5)$.

Using the Quotient Rule: ${\log }_{10}(1000)-{\log }_{10}(10)-{\log }_{10}(5)={\log }_{10}\left(\frac{1000}{10\cdot 5}\right)$

Now, simplify the quotient: ${\log }_{10}\left(\frac{1000}{10\cdot 5}\right)={\log }_{10}\left(\frac{1000}{50}\right)={\log }_{10}(20)$

So, the condensed expression is: ${\log }_{10}(1000)-{\log }_{10}(10)-{\log }_{10}(5)={\log }_{10}(20)$.

Example 3: Condense $2\cdot {\log }_3(27)+3\cdot {\log }_3(9)+{\log }_3(81)$.

Using the Power Rule and the Product Rule: $2\cdot {\log }_3(27)+3\cdot {\log }_3(9)+{\log }_3(81)={\log }_3(27^2)+{\log }_3(9^3)+{\log }_3(81)$

Now, simplify the products: ${\log }_3(27^2)={\log }_3(729)$ ${\log }_3(9^3)={\log }_3(729)$

So, the condensed expression is: $2\cdot {\log }_3(27)+3\cdot {\log }_3(9)+{\log }_3(81)={\log }_3(729)+{\log }_3(729)+{\log }_3(81)$.

By applying the appropriate logarithmic properties, you can condense logarithmic expressions and simplify them into a more compact form. This can be particularly useful for simplifying mathematical expressions and equations.

The Change-of-Base Formula for logarithms is a useful tool when you want to change the base of a logarithm from one value to another. It is especially helpful when dealing with logarithms that don't have convenient integer bases. The formula is as follows:

If you have a logarithm ${\log }_{�}(�)$ and want to change the base to $�$, you can use the Change-of-Base Formula:

${\log }_{�}(�)=\frac{{\log }_{�}(�)}{{\log }_{�}(�)}$

Here's how to use the Change-of-Base Formula with some examples:

Example 1: Change ${\log }_2(16)$ to a base-10 logarithm using the Change-of-Base Formula:

${\log }_2(16)=\frac{{\log }_{10}(16)}{{\log }_{10}(2)}$

Now, you can calculate the values on the right side using base-10 logarithms. Typically, a calculator with a base-10 logarithm function can be used for this:

$\frac{{\log }_{10}(16)}{{\log }_{10}(2)}\approx \frac{1.2041}{0.3010}\approx 4$

So, ${\log }_2(16)\approx 4$ when calculated using base-10 logarithms.

Example 2: Change ${\log }_5(25)$ to a natural logarithm (base $�$) using the Change-of-Base Formula:

${\log }_5(25)=\frac{\ln (25)}{\ln (5)}$

Now, you can calculate the values on the right side using natural logarithms (base $�$):

$\frac{\ln (25)}{\ln (5)}\approx \frac{3.2189}{1.6094}\approx 2$

So, ${\log }_5(25)\approx 2$ when calculated using natural logarithms.

The Change-of-Base Formula allows you to change the base of a logarithm to one that's more convenient for your calculations or when you need to use a particular base, like the common logarithm (base 10) or natural logarithm (base $�$).

For the following exercises, condense to a single logarithm if possible. 9. ln(4)+2n(x)+4n(y) 10. log3(1)+log5(a)+log5(18)+log5(b) 11. logb(25)−logb(8) 12. ln(a)−ln(d)−ln(c) 13. −logb(19) 14. 16ln(7)

Let's condense the given expressions to a single logarithm where possible:

9. $\ln (4)+2�(�)+4�(�)$

We can condense this expression by using the properties of logarithms. Remember that the sum of logarithms is equivalent to the logarithm of their product. So:

$\ln (4)+2�(�)+4�(�)=\ln (4)+\ln ({�}^{2�})+\ln ({�}^{4�})$

Now, using the Product Rule, we can combine the logarithms:

$\ln (4{�}^{2�}{�}^{4�})$

10. ${\log }_3(1)+{\log }_5(�)+{\log }_5(18)+{\log }_5(�)$

The logarithm of 1 is always 0, so we can simplify the expression:

${\log }_3(1)+{\log }_5(�)+{\log }_5(18)+{\log }_5(�)=0+{\log }_5(�)+{\log }_5(18)+{\log }_5(�)$

Now, using the Product Rule, we can combine the logarithms:

${\log }_5(��\cdot 18)$

11. ${\log }_{�}(25)-{\log }_{�}(8)$

Using the Quotient Rule for logarithms, we can combine these logarithms:

${\log }_{�}\left(\frac{25}{8}\right)$

12. $\ln (�)-\ln (�)-\ln (�)$

Using the Quotient Rule for logarithms, we can combine these logarithms:

$\ln \left(\frac{�}{�}\right)-\ln (�)$

13. $-{\log }_{�}(19)$

The negative sign in front of the logarithm is a power, which can be used to rewrite the logarithm as a reciprocal:

$-{\log }_{�}(19)={\log }_{�}\left(\frac{1}{19}\right)$

14. $16\ln (7)$

We can use the Power Rule for logarithms to move the exponent as a coefficient:

$\ln (7^{16})$

So, these expressions have been condensed to single logarithms where possible.