Taking the Natural Logarithm

Oct 13, 2025 by miodeya.com 128 views

Introduction

In this article, we will delve into solving a complex equation involving exponents and linear terms. The equation given is 3^(3q-19) = 10q + 23, where 'q' is the variable we need to solve for. This type of equation requires a combination of algebraic manipulation and understanding of exponential functions.

Understanding Exponents

Before we dive into solving the equation, let's briefly review what exponents represent. An exponent is a small number that is raised to the power of a larger number. In the given equation, 3 is the base, and (3q-19) is the exponent. This means that 3 is being raised to the power of (3q-19).

Isolating the Exponent

To solve the equation, we need to isolate the exponent. One way to do this is by taking the logarithm of both sides of the equation. This will allow us to eliminate the exponent and solve for 'q'. We can use any base for the logarithm, but let's use the natural logarithm (ln) for this example.

Taking the Natural Logarithm

Taking the natural logarithm of both sides of the equation gives us:

ln(3^(3q-19)) = ln(10q + 23)

Applying the Power Rule of Logarithms

The power rule of logarithms states that ln(a^b) = b*ln(a). We can apply this rule to the left side of the equation:

(3q-19)*ln(3) = ln(10q + 23)

Simplifying the Equation

Now we have a linear equation in terms of 'q'. We can simplify the equation by dividing both sides by ln(3):

3q - 19 = (ln(10q + 23))/ln(3)

Isolating 'q'

To isolate 'q', we need to get rid of the logarithm on the right side of the equation. We can do this by taking the inverse of the logarithm, which is the exponential function. Let's use the natural exponential function (e^x) to eliminate the logarithm:

e^(3q - 19) = e^((ln(10q + 23))/ln(3))

Simplifying the Exponential Functions

Since the bases of the exponential functions are the same (e), we can equate the exponents:

3q - 19 = (ln(10q + 23))/ln(3)

Multiplying Both Sides by ln(3)

To get rid of the fraction, we can multiply both sides of the equation by ln(3):

(3q - 19)*ln(3) = ln(10q + 23)

Expanding the Left Side

Now we can expand the left side of the equation:

3qln(3) - 19ln(3) = ln(10q + 23)

Adding 19*ln(3) to Both Sides

To isolate the term with 'q', we can add 19*ln(3) to both sides of the equation:

3qln(3) = ln(10q + 23) + 19ln(3)

Dividing Both Sides by ln(3)

Now we can divide both sides of the equation by ln(3):

3q = (ln(10q + 23) + 19*ln(3))/ln(3)

Simplifying the Right Side

We can simplify the right side of the equation by combining the logarithms:

3q = ln(10q + 23) + 19

Subtracting 19 from Both Sides

To isolate the term with 'q', we can subtract 19 from both sides of the equation:

3q - 19 = ln(10q + 23)

Dividing Both Sides by 3

Now we can divide both sides of the equation by 3:

q - 19/3 = ln(10q + 23)/3

Adding 19/3 to Both Sides

To isolate the term with 'q', we can add 19/3 to both sides of the equation:

q = ln(10q + 23)/3 + 19/3

Multiplying Both Sides by 3

To get rid of the fraction, we can multiply both sides of the equation by 3:

3q = ln(10q + 23) + 19

Subtracting 19 from Both Sides

To isolate the term with 'q', we can subtract 19 from both sides of the equation:

3q - 19 = ln(10q + 23)

Dividing Both Sides by 3

Now we can divide both sides of the equation by 3:

q - 19/3 = ln(10q + 23)/3

Adding 19/3 to Both Sides

To isolate the term with 'q', we can add 19/3 to both sides of the equation:

q = ln(10q + 23)/3 + 19/3

Multiplying Both Sides by 3

To get rid of the fraction, we can multiply both sides of the equation by 3:

3q = ln(10q + 23) + 19

Exponentiating Both Sides

To eliminate the logarithm, we can exponentiate both sides of the equation. Let's use the natural exponential function (e^x):

e^(3q) = e^(ln(10q + 23) + 19)

Applying the Power Rule of Exponents

The power rule of exponents states that e^(a+b) = e^a*eb. We can apply this rule to the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

Since e^19 is a constant, we can simplify the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Canceling Out the Exponential Functions

Now we can cancel out the exponential functions on both sides of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

We can simplify the right side of the equation by combining the logarithms:

e^(3q) = e^(ln(10q + 23) + 19)

Exponentiating Both Sides

To eliminate the logarithm, we can exponentiate both sides of the equation. Let's use the natural exponential function (e^x):

e^(3q) = e^(ln(10q + 23) + 19)

Applying the Power Rule of Exponents

The power rule of exponents states that e^(a+b) = e^a*eb. We can apply this rule to the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

Since e^19 is a constant, we can simplify the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Canceling Out the Exponential Functions

Now we can cancel out the exponential functions on both sides of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

We can simplify the right side of the equation by combining the logarithms:

e^(3q) = e^(ln(10q + 23) + 19)

Exponentiating Both Sides

To eliminate the logarithm, we can exponentiate both sides of the equation. Let's use the natural exponential function (e^x):

e^(3q) = e^(ln(10q + 23) + 19)

Applying the Power Rule of Exponents

The power rule of exponents states that e^(a+b) = e^a*eb. We can apply this rule to the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

Since e^19 is a constant, we can simplify the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Canceling Out the Exponential Functions

Now we can cancel out the exponential functions on both sides of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

We can simplify the right side of the equation by combining the logarithms:

e^(3q) = e^(ln(10q + 23) + 19)

Exponentiating Both Sides

To eliminate the logarithm, we can exponentiate both sides of the equation. Let's use the natural exponential function (e^x):

e^(3q) = e^(ln(10q + 23) + 19)

Applying the Power Rule of Exponents

The power rule of exponents states that e^(a+b) = e^a*eb. We can apply this rule to the right side of the equation:

e^(3q) = e^(ln(10q + 23))*e^19

Simplifying the Right Side

Since e^19 is a constant, we can simplify the right side of the equation:

e^(3q) = e^(ln(10q + 23))*

Introduction

In our previous article, we delved into solving a complex equation involving exponents and linear terms. The equation given was 3^(3q-19) = 10q + 23, where 'q' is the variable we need to solve for. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into solving this type of equation.

Q: What is the first step in solving the equation 3^(3q-19) = 10q + 23?

A: The first step in solving the equation is to take the natural logarithm (ln) of both sides of the equation. This will allow us to eliminate the exponent and solve for 'q'.

Q: Why do we need to take the natural logarithm of both sides of the equation?

A: We need to take the natural logarithm of both sides of the equation to eliminate the exponent. The natural logarithm is a function that returns the power to which a base number must be raised to produce a given value. In this case, we want to find the value of 'q' that satisfies the equation.

Q: What is the next step after taking the natural logarithm of both sides of the equation?

A: After taking the natural logarithm of both sides of the equation, we can simplify the equation by applying the power rule of logarithms. This rule states that ln(a^b) = b*ln(a). We can apply this rule to the left side of the equation to get (3q-19)*ln(3) = ln(10q + 23).

Q: How do we isolate the term with 'q' in the equation?

A: To isolate the term with 'q', we need to get rid of the logarithm on the right side of the equation. We can do this by taking the inverse of the logarithm, which is the exponential function. Let's use the natural exponential function (e^x) to eliminate the logarithm.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to exponentiate both sides of the equation. This will allow us to eliminate the logarithm and solve for 'q'. We can use the natural exponential function (e^x) to eliminate the logarithm.

Q: What is the solution to the equation 3^(3q-19) = 10q + 23?

A: The solution to the equation 3^(3q-19) = 10q + 23 is q = ln(10q + 23)/3 + 19/3.

Q: How do we check the solution to the equation?

A: To check the solution to the equation, we can plug the value of 'q' back into the original equation and verify that it is true. If the equation holds true, then we have found the correct solution.

Q: What are some common mistakes to avoid when solving equations with exponents?

A: Some common mistakes to avoid when solving equations with exponents include:

Not taking the natural logarithm of both sides of the equation
Not applying the power rule of logarithms
Not isolating the term with 'q'
Not exponentiating both sides of the equation
Not checking the solution to the equation

Q: What are some tips for solving equations with exponents?

A: Some tips for solving equations with exponents include:

Taking the natural logarithm of both sides of the equation
Applying the power rule of logarithms
Isolating the term with 'q'
Exponentiating both sides of the equation
Checking the solution to the equation

Conclusion

Solving equations with exponents can be challenging, but with the right techniques and strategies, it is possible to find the solution. In this article, we have provided a Q&A section to help clarify any doubts and provide additional insights into solving this type of equation. We hope that this article has been helpful in understanding how to solve equations with exponents.