MTH120 College Algebra Chapter 6.3

 6.3 Logarithmic Functions

Converting from logarithmic to exponential form involves changing an equation or expression written in logarithmic notation into an equivalent exponential form. The general forms of logarithmic and exponential equations are as follows:

Logarithmic Form: ${\log }_{�}(�)=�$

In this equation:

• ${\log }_{�}$ represents the logarithm with base $�$.
• $�$ is the value you are taking the logarithm of.
• $�$ is the result of the logarithm.

Exponential Form: ${�}^{�}=�$

In this equation:

• $�$ is the base.
• $�$ is the exponent.
• $�$ is the result of the exponentiation.

To convert from logarithmic to exponential form, you need to identify the base ($�$), the exponent ($�$), and the result of the logarithm ($�$). Here's how you do it step by step:

1. Start with the given logarithmic equation or expression in the form ${\log }_{�}(�)=�$.

2. Identify the base ($�$). This is the same base that will be used in the equivalent exponential equation.

3. Identify the exponent ($�$). This is the value to which the base ($�$) will be raised in the exponential form.

4. Identify the result of the logarithm ($�$). This is the value that results from taking the logarithm of a number.

5. Write the equivalent exponential equation by placing the base ($�$), the exponent ($�$), and the result ($�$) in the exponential form ${�}^{�}=�$.

Let's illustrate this with an example:

Example: Convert the logarithmic equation ${\log }_2(8)=3$ to exponential form.

1. Start with the given logarithmic equation: ${\log }_2(8)=3$.

2. Identify the base ($�$): In this case, the base is 2.

3. Identify the exponent ($�$): The exponent is 3.

4. Identify the result of the logarithm ($�$): The result of the logarithm is 8.

5. Write the equivalent exponential equation: $2^3=8$.

So, the exponential form of the logarithmic equation ${\log }_2(8)=3$ is $2^3=8$.

The logarithmic function, often denoted as ${\log }_{�}(�)$, is the inverse operation of exponentiation. It is used to find the exponent ($�$) to which a specific base ($�$) must be raised in order to obtain a given number ($�$). In mathematical terms, the logarithmic function can be defined as:

${\log }_{�}(�)=�$

Where:

• ${\log }_{�}$ represents the logarithm with base $�$.
• $�$ is the number for which you want to find the exponent.
• $�$ is the exponent that, when used as the power of $�$, equals $�$.

Here's an example to illustrate the use of the logarithmic function:

Example: Find the logarithm base 10 of 100.

To find ${\log }_{10}(100)$, you are looking for the exponent ($�$) to which the base 10 must be raised to obtain the number 100. In other words, you want to find $�$ in the equation:

$10^{�}=100$

To solve this equation, you can use the logarithmic function. Taking the base 10 logarithm of both sides, you get:

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

Since ${\log }_{10}(10^{�})$ is equivalent to $�$ (the logarithm undoes exponentiation when using the same base), you have:

$�={\log }_{10}(100)$

Now, calculate ${\log }_{10}(100)$ to find $�$. In this case, $�$ represents the exponent to which 10 must be raised to equal 100. The answer is $�=2$, because $10^2=100$.

So, ${\log }_{10}(100)=2$. The logarithmic function allows you to find the exponent needed to achieve a specific value using a given base, and it is commonly used in various fields, including mathematics, science, engineering, and finance.

Logarithmic functions are widely used in engineering, particularly in the fields of signal processing and control systems. One common application is in measuring the decibel (dB) level of a signal, which quantifies the relative intensity or power of a signal. The formula to calculate the decibel level is based on a logarithmic function.

Example: Calculating Decibel (dB) Levels in Engineering

In engineering, the decibel scale is commonly used to express the intensity or power ratio of signals, such as sound, electrical power, or signal amplitudes. The decibel level ($�$) is calculated using the following formula:

$�=10\cdot {\log }_{10}\left(\frac{�}{{�}_0}\right)$

Where:

• $�$ is the decibel level.
• $�$ is the power or intensity of the signal of interest.
• ${�}_0$ is a reference power level (often a standard or threshold value).

Here's an example:

Problem: Calculate the decibel level of a sound signal with a power of 10 milliwatts ($�=10$ mW) relative to a reference power level of 1 milliwatt (${�}_0=1$ mW).

Solution:

Using the formula, we can calculate the decibel level ($�$) as follows:

$�=10\cdot {\log }_{10}\left(\frac{10\text{mW}}{1\text{mW}}\right)$

Simplify the expression inside the logarithm:

$�=10\cdot {\log }_{10}(10)$

Now, calculate the logarithm:

$�=10\cdot 1=10\text{dB}$

So, the sound signal with a power of 10 milliwatts relative to a reference power level of 1 milliwatt has a decibel level of 10 dB.

In this engineering example, the logarithmic function is used to express the power ratio between two signals on a logarithmic scale, making it easier to compare and work with signals of varying intensities, such as in audio engineering, telecommunications, and electrical engineering.

To convert an equation from logarithmic form, ${\log }_{�}(�)=�$, to exponential form, you can follow these steps:

Logarithmic Form: ${\log }_{�}(�)=�$

Exponential Form: ${�}^{�}=�$

Here's how you convert from logarithmic to exponential form:

1. Start with the given logarithmic equation in the form ${\log }_{�}(�)=�$.

2. Identify the base ($�$) from the logarithmic equation. This base will remain the same in the exponential equation.

3. Identify the result ($�$) of the logarithm, which is on the right side of the equation.

4. Identify the exponent ($�$) to which the base ($�$) must be raised to obtain $�$.

5. Write the equivalent exponential equation by placing the base ($�$), the exponent ($�$), and the result ($�$) in the exponential form ${�}^{�}=�$.

Here's an example:

Example: Convert the logarithmic equation ${\log }_2(8)=3$ to exponential form.

1. Start with the given logarithmic equation: ${\log }_2(8)=3$.

2. Identify the base ($�$): In this case, the base is 2.

3. Identify the result ($�$): The result of the logarithm is 8.

4. Identify the exponent ($�$): The exponent is 3.

5. Write the equivalent exponential equation: $2^3=8$.

So, the exponential form of the logarithmic equation ${\log }_2(8)=3$ is $2^3=8$.

Converting logarithmic equations to exponential form is a fundamental skill when working with logarithmic functions, as it allows you to solve equations and understand relationships between numbers and their logarithms.

Evaluating logarithms involves finding the numerical value of a logarithmic expression. Logarithms are often used to solve equations or express relationships involving exponentiation in a more manageable form. To evaluate a logarithm, you typically use a base and an argument. Here's how to evaluate logarithms:

1. Understand the Logarithmic Notation:

   • A logarithm, written as ${\log }_{�}(�)$, consists of three parts:
     • The base ($�$): This is the number you are using as the base of the logarithm.
     • The argument ($�$): This is the number you want to find the logarithm of.
     • The result: The value you are trying to calculate.

2. Use the Appropriate Base:

   • The base of the logarithm determines which number you are using as the base for the exponentiation. Common bases include 10 (common logarithm, often written as $\log (�)$), 2, and the natural logarithm base $�$, written as $\ln (�)$.

3. Calculate the Logarithm:

   • Use the base and argument to calculate the logarithm. The formula is: ${\log }_{�}(�)=�$ This equation tells you that ${�}^{�}=�$, where $�$ is the result of the logarithm.

4. Use a Calculator or Table (if needed):

   • Most calculators have logarithmic functions built in. To evaluate a logarithm, simply enter the base and the argument. For example, to calculate ${\log }_{10}(100)$, you would enter $\log (100)$ on a calculator.

Here are some examples of evaluating logarithms:

Example 1: Evaluate ${\log }_{10}(100)$.

• Using base 10 and argument 100, we want to find the exponent ($�$) that makes $10^{�}=100$.
• By observation, we know that $10^2=100$.
• So, ${\log }_{10}(100)=2$.

Example 2: Evaluate $\ln ({�}^3)$.

• Here, we're using the natural logarithm base $�$, and the argument is ${�}^3$.
• The logarithmic form is $\ln ({�}^3)=3$.

Example 3: Evaluate ${\log }_2(8)$.

• Using base 2 and argument 8, we want to find the exponent ($�$) that makes $2^{�}=8$.
• By observation, we know that $2^3=8$.
• So, ${\log }_2(8)=3$.

Example 4: Evaluate ${\log }_{10}(1)$.

• When the argument is 1, the result is always 0, regardless of the base.
• So, ${\log }_{10}(1)=0$.

Example 5: Evaluate ${\log }_5(25)$.

• Using base 5 and argument 25, we want to find the exponent ($�$) that makes $5^{�}=25$.
• By observation, we know that $5^2=25$.
• So, ${\log }_5(25)=2$.

These examples illustrate how to evaluate logarithms by understanding the relationship between the base, argument, and result. Evaluating logarithms is a fundamental skill when working with exponential and logarithmic functions.

Common logarithms, often denoted as $\log (�)$ without a specified base, have a base of 10. They are a type of logarithm commonly used in mathematics, science, and engineering. To use common logarithms, you can follow these steps:

1. Understand the Common Logarithmic Notation:

   • Common logarithms are logarithms with a base of 10. They are often written as $\log (�)$ or ${\log }_{10}(�)$, and the base is understood to be 10.

2. Use Common Logarithms for Base-10 Calculations:

   • Common logarithms are particularly useful when you need to work with numbers in base-10, such as decimal numbers.

3. Evaluate Common Logarithms:

   • To evaluate a common logarithm, simply take the logarithm of the argument (the number inside the logarithm).

Example 1: Evaluate $\log (100)$.

• This is a common logarithm, which is a base-10 logarithm.
• You want to find the exponent ($�$) that makes $10^{�}=100$.
• By observation, we know that $10^2=100$.
• So, $\log (100)=2$.

Example 2: Solve for $�$ in the equation $\log (�)=3$.

• You have a common logarithm $\log (�)$ equal to 3.
• This means you're looking for a number ($�$) such that $10^3=�$.
• So, $�=1000$.

Example 3: Evaluate $\log (1)$.

• When the argument of a common logarithm is 1, the result is always 0 because $10^0=1$.
• So, $\log (1)=0$.

Example 4: Evaluate $\log (0.1)$.

• In this case, you're evaluating a common logarithm of a decimal number.
• The result is $\log (0.1)=-1$ because $10^{-1}=0.1$.

Common logarithms are a convenient way to work with base-10 numbers and are often used when dealing with quantities that have orders of magnitude. They are an essential tool in scientific notation, where they simplify the expression of very large or very small numbers.

Natural logarithms, often denoted as $\ln (�)$, are logarithms with a base of $�$, where $�$ is Euler's number, approximately equal to 2.71828. Natural logarithms are widely used in mathematics, science, and engineering. To use natural logarithms, you can follow these steps:

1. Understand the Natural Logarithmic Notation:

   • Natural logarithms are logarithms with a base of $�$, denoted as $\ln (�)$.
   • The number $�$ is an irrational constant, and it is approximately 2.71828.

2. Use Natural Logarithms for $�$-Based Calculations:

   • Natural logarithms are particularly useful when you need to work with exponential functions based on $�$.

3. Evaluate Natural Logarithms:

   • To evaluate a natural logarithm, simply take the logarithm of the argument (the number inside the logarithm).

Example 1: Evaluate $\ln ({�}^3)$.

• In this case, you're evaluating a natural logarithm of an exponential function.
• The natural logarithm "undoes" the exponential function, so $\ln ({�}^3)=3$.

Example 2: Solve for $�$ in the equation $\ln (�)=2$.

• You have a natural logarithm $\ln (�)$ equal to 2.
• This means you're looking for a number ($�$) such that ${�}^2=�$.
• So, $�={�}^2$, which is approximately 7.38906.

Example 3: Evaluate $\ln (1)$.

• When the argument of a natural logarithm is 1, the result is always 0 because ${�}^0=1$.
• So, $\ln (1)=0$.

Example 4: Evaluate $\ln (0.1)$.

• In this case, you're evaluating a natural logarithm of a decimal number.
• The result is $\ln (0.1)\approx -2.30259$ because ${�}^{-2.30259}\approx 0.1$.

Natural logarithms are particularly important when dealing with exponential growth or decay, as well as in calculus when working with derivatives and integrals involving exponential functions. They are commonly used in various scientific and mathematical applications, including finance, physics, and engineering.

For the following exercises, solve for x by converting the logarithmic equation to exponential form. 1. log3(x)=3 2. log2(x)=−2 3. log5(x)=1 4. log3(x)=5 5. log2(x)=7 6. log9(x)=15 7. log18(x)=1 8. log6(x)=−9 9. log(x)=8 10. ln(x)=12

To solve for $�$ in each of the given logarithmic equations, you can convert them to exponential form and then solve for $�$. Here are the solutions:

1. ${\log }_3(�)=3$ can be converted to exponential form as $3^3=�$. Therefore, $�=27$.

2. ${\log }_2(�)=-2$ can be converted to exponential form as $2^{-2}=�$. Therefore, $�=\frac{1}{4}$.

3. ${\log }_5(�)=1$ can be converted to exponential form as $5^1=�$. Therefore, $�=5$.

4. ${\log }_3(�)=5$ can be converted to exponential form as $3^5=�$. Therefore, $�=243$.

5. ${\log }_2(�)=7$ can be converted to exponential form as $2^7=�$. Therefore, $�=128$.

6. ${\log }_9(�)=15$ can be converted to exponential form as $9^{15}=�$.

7. ${\log }_{18}(�)=1$ can be converted to exponential form as $18^1=�$. Therefore, $�=18$.

8. ${\log }_6(�)=-9$ can be converted to exponential form as $6^{-9}=�$. Therefore, $�=\frac{1}{6^9}$.

9. $\log (�)=8$ (assuming base 10 for the common logarithm) can be converted to exponential form as $10^8=�$. Therefore, $�=100,000,000$.

10. $\ln (�)=12$ can be converted to exponential form as ${�}^{12}=�$. Therefore, $�\approx 162754.79$ (rounded to approximately five decimal places).

These are the solutions for $�$ in each of the given logarithmic equations after converting them to exponential form.