Solving Logarithmic Equations Log(x) + Log(3) = Log(18)

Solve the equation log(x) + log(3) = log(18) for x.

In this comprehensive guide, we will delve into the intricacies of solving logarithmic equations, focusing specifically on the equation ${ \log x + \log 3 = \log 18 }$. Logarithmic equations are a fundamental topic in mathematics, appearing in various fields such as calculus, physics, and engineering. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. This article aims to provide a detailed, step-by-step approach to solving the given equation, while also discussing the underlying principles and properties of logarithms that make this solution possible. Whether you are a student grappling with logarithmic equations for the first time or someone looking to refresh your knowledge, this guide will provide the clarity and understanding you need.

Understanding Logarithms

Before we tackle the equation ${ \log x + \log 3 = \log 18 }$, it is essential to have a solid grasp of what logarithms are and how they behave. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, if we have ${ b^y = x }$, then the logarithm of ${ x }$ to the base ${ b }$ is written as ${ \log_b x = y }$. Here, ${ b }$ is the base, ${ x }$ is the argument, and ${ y }$ is the logarithm. For instance, ${ 10^2 = 100 }$ can be rewritten in logarithmic form as ${ \log_{10} 100 = 2 }$.

When no base is explicitly written, as in our equation ${ \log x + \log 3 = \log 18 }$, it is generally understood that the base is 10. This is known as the common logarithm. Therefore, ${ \log x }$ is the same as ${ \log_{10} x }$. Another important type of logarithm is the natural logarithm, denoted by ${ \ln x }$, which has a base of ${ e }$ (Euler's number, approximately 2.71828). Understanding these basics is crucial because the properties of logarithms allow us to simplify complex expressions and solve equations more easily. The ability to convert between exponential and logarithmic forms is a cornerstone of working with logarithms effectively.

Key Properties of Logarithms

To effectively solve logarithmic equations, it's crucial to understand and apply the properties of logarithms. These properties allow us to manipulate and simplify logarithmic expressions, making it easier to isolate variables and find solutions. Here are some of the fundamental properties that we will use in solving our equation, ${ \log x + \log 3 = \log 18 }$:

1. Product Rule: The logarithm of the product of two numbers is equal to the sum of their logarithms. Mathematically, this is expressed as ${ \log_b(mn) = \log_b m + \log_b n }$. This property is particularly useful for combining logarithmic terms, as we'll see in our solution. For example, ${ \log(2 imes 5) }$ is the same as ${ \log 2 + \log 5 }$.

2. Quotient Rule: The logarithm of the quotient of two numbers is equal to the difference of their logarithms. This is represented as ${ \log_b(\frac{m}{n}) = \log_b m - \log_b n }$. This property is the counterpart to the product rule and helps in simplifying expressions involving division. For example, ${ \log(\frac{10}{2}) }$ is the same as ${ \log 10 - \log 2 }$.

3. Power Rule: The logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number. This is written as ${ \log_b(m^p) = p \log_b m }$. The power rule is invaluable for dealing with exponents within logarithms. For example, ${ \log(2^3) }$ is the same as ${ 3 \log 2 }$.

4. Change of Base Rule: This rule allows us to convert logarithms from one base to another. It states that ${ \log_b a = \frac{\log_c a}{\log_c b} }$, where ${ c }$ is a new base. This is particularly useful when dealing with logarithms that don't have a common base like 10 or ${ e }$. For example, if you need to find ${ \log_2 7 }$ and your calculator only computes base 10 logarithms, you can use this rule to convert it to ${ \frac{\log_{10} 7}{\log_{10} 2} }$.

5. Logarithm of 1: The logarithm of 1 to any base is always 0. This is because any number raised to the power of 0 is 1. So, ${ \log_b 1 = 0 }$ for any base ${ b }$. For example, ${ \log_{10} 1 = 0 }$ and ${ \ln 1 = 0 }$.

6. Logarithm of the Base: The logarithm of a number to the same base is always 1. This is because any number raised to the power of 1 is itself. So, ${ \log_b b = 1 }$ for any base ${ b }$. For example, ${ \log_{10} 10 = 1 }$ and ${ \ln e = 1 }$.

These properties are the tools we will use to simplify and solve logarithmic equations. Understanding and being able to apply them is essential for success in this area of mathematics.

Step-by-Step Solution of ${ \log x + \log 3 = \log 18 }$

Now, let's apply our understanding of logarithms to solve the equation ${ \log x + \log 3 = \log 18 }$. We will proceed step-by-step, explaining each operation and the logarithmic properties used.

Step 1: Applying the Product Rule

The first step in solving this equation is to recognize that we have a sum of logarithms on the left side. According to the product rule of logarithms, the sum of the logarithms of two numbers is equal to the logarithm of their product. In mathematical terms, ${ \log_b m + \log_b n = \log_b(mn) }$. Applying this rule to our equation, we can combine the two logarithmic terms on the left side:

${ \log x + \log 3 = \log(x imes 3) }$

So, our equation now becomes:

${ \log(3x) = \log 18 }$

This step simplifies the equation by reducing two logarithmic terms into one, making it easier to handle.

Step 2: Eliminating Logarithms

Now that we have a single logarithm on each side of the equation, we can eliminate the logarithms by using the fact that if ${ \log_b m = \log_b n }$, then ${ m = n }$. In other words, if the logarithms of two expressions are equal, then the expressions themselves must be equal. Applying this to our equation ${ \log(3x) = \log 18 }$, we can equate the arguments of the logarithms:

${ 3x = 18 }$

This step is crucial because it transforms the logarithmic equation into a simple algebraic equation, which is much easier to solve.

Step 3: Solving for ${ x }$

We now have a straightforward linear equation: ${ 3x = 18 }$. To solve for ${ x }$, we need to isolate ${ x }$ by dividing both sides of the equation by 3:

${ \frac{3x}{3} = \frac{18}{3} }$

This simplifies to:

${ x = 6 }$

Thus, we have found the value of ${ x }$ that satisfies the equation.

Step 4: Verifying the Solution

It is always a good practice to verify the solution by substituting the value of ${ x }$ back into the original equation to ensure it holds true. Our original equation was ${ \log x + \log 3 = \log 18 }$. Substituting ${ x = 6 }$, we get:

${ \log 6 + \log 3 = \log 18 }$

Using the product rule of logarithms, we can combine the left side:

${ \log(6 imes 3) = \log 18 }$

${ \log 18 = \log 18 }$

Since the left side equals the right side, our solution ${ x = 6 }$ is correct.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and arrive at the correct solution more efficiently. Here are some common mistakes to watch out for:

1. Incorrectly Applying Logarithmic Properties: The properties of logarithms are powerful tools, but they must be applied correctly. A common mistake is to misapply the product, quotient, or power rules. For example, students might incorrectly assume that ${ \log(m + n) }$ is equal to ${ \log m + \log n }$, which is not true. Remember, ${ \log(mn) = \log m + \log n }$, not ${ \log(m + n) }$. Similarly, ensure you understand when and how to apply the quotient and power rules.

2. Ignoring the Domain of Logarithms: Logarithms are only defined for positive arguments. This means that the argument of a logarithm (the value inside the logarithm) must be greater than zero. When solving logarithmic equations, it's crucial to check that the solutions you find do not result in taking the logarithm of a negative number or zero. For instance, if you find a solution ${ x = -2 }$ for an equation involving ${ \log x }$, this solution is extraneous because ${ \log(-2) }$ is undefined.

3. Incorrectly Eliminating Logarithms: When you have a logarithmic equation of the form ${ \log_b m = \log_b n }$, you can equate the arguments ${ m }$ and ${ n }$. However, it's essential to ensure that both sides of the equation are single logarithmic terms before eliminating the logarithms. A common mistake is to try to eliminate logarithms when there are additional terms outside the logarithms. For example, in the equation ${ \log x + 2 = \log 5 }$, you cannot simply equate ${ x }$ and 5. You must first isolate the logarithmic terms.

4. Forgetting to Verify Solutions: As we demonstrated in our step-by-step solution, it's crucial to verify your solutions by plugging them back into the original equation. This is particularly important for logarithmic equations because extraneous solutions can arise due to the domain restrictions of logarithms. Verification ensures that the solutions you find are valid.

5. Misunderstanding the Base: Always pay attention to the base of the logarithm. If no base is written, it is usually assumed to be 10 (common logarithm). However, if the base is different, such as the natural logarithm (base ${ e }$), you must use the appropriate properties and rules for that base. Confusing the base can lead to incorrect calculations and solutions.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving logarithmic equations.

Real-World Applications of Logarithms

Logarithms are not just abstract mathematical concepts; they have numerous practical applications in various fields of science, engineering, and everyday life. Understanding these applications can help you appreciate the significance of logarithms and their role in solving real-world problems. Here are some notable examples:

1. Decibel Scale (Acoustics): The decibel (dB) scale, used to measure sound intensity, is based on logarithms. The loudness of a sound is perceived logarithmically by the human ear, meaning that a small change in decibels corresponds to a large change in sound intensity. The formula for calculating decibels is ${ dB = 10 \log_{10}(\frac{I}{I_0}) }$, where ${ I }$ is the sound intensity and ${ I_0 }$ is a reference intensity. This logarithmic scale allows us to represent a wide range of sound intensities in a manageable way.

2. Richter Scale (Seismology): The Richter scale, used to measure the magnitude of earthquakes, is another logarithmic scale. The magnitude ${ M }$ of an earthquake is calculated using the formula ${ M = \log_{10}(\frac{A}{A_0}) }$, where ${ A }$ is the amplitude of the seismic waves and ${ A_0 }$ is a reference amplitude. Each whole number increase on the Richter scale represents a tenfold increase in the amplitude of the seismic waves and a roughly 31.6 times increase in the energy released. This logarithmic scale is essential for quantifying and comparing the sizes of earthquakes.

3. pH Scale (Chemistry): In chemistry, the pH scale is used to measure the acidity or alkalinity of a solution. The pH is defined as the negative logarithm of the hydrogen ion concentration: ${ pH = -\log_{10}[H^+] }$, where ${ [H^+] }$ is the concentration of hydrogen ions in moles per liter. A pH of 7 is neutral, values less than 7 are acidic, and values greater than 7 are alkaline. The logarithmic nature of the pH scale means that each whole number change in pH represents a tenfold change in hydrogen ion concentration.

4. Exponential Growth and Decay: Logarithms are crucial in modeling phenomena that exhibit exponential growth or decay. For example, population growth, radioactive decay, and compound interest calculations often involve exponential functions, and logarithms are used to solve for variables in these models. The general formula for exponential growth or decay is ${ N(t) = N_0 e^{kt} }$, where ${ N(t) }$ is the quantity at time ${ t }$, ${ N_0 }$ is the initial quantity, ${ k }$ is the rate of growth or decay, and ${ e }$ is the base of the natural logarithm. Logarithms are used to solve for ${ t }$ or ${ k }$ in these equations.

5. Computer Science: Logarithms are fundamental in computer science, particularly in the analysis of algorithms. The time complexity of many algorithms is expressed using logarithmic functions. For example, the time complexity of binary search is ${ O(\log n) }$, where ${ n }$ is the number of elements. Logarithms also appear in data structures such as binary trees and heaps, where logarithmic relationships determine the efficiency of operations.

6. Finance: In finance, logarithms are used to calculate compound interest, rates of return, and the time it takes for investments to grow. The compound interest formula, ${ A = P(1 + \frac{r}{n})^{nt} }$, can be solved for ${ t }$ using logarithms, where ${ A }$ is the final amount, ${ P }$ is the principal, ${ r }$ is the annual interest rate, ${ n }$ is the number of times interest is compounded per year, and ${ t }$ is the time in years.

These examples illustrate the wide range of applications of logarithms in various fields. By understanding logarithms, you gain a powerful tool for solving problems and making sense of the world around you.

Conclusion

In this guide, we have provided a comprehensive explanation of how to solve the logarithmic equation ${ \log x + \log 3 = \log 18 }$. We began by discussing the fundamental properties of logarithms, including the product rule, which is essential for simplifying the equation. We then walked through a step-by-step solution, demonstrating how to combine logarithmic terms, eliminate logarithms, and solve for the variable ${ x }$. We also emphasized the importance of verifying the solution to ensure its validity.

Furthermore, we highlighted common mistakes to avoid when solving logarithmic equations, such as incorrectly applying logarithmic properties, ignoring the domain of logarithms, and forgetting to verify solutions. By being aware of these pitfalls, you can improve your accuracy and avoid errors. Finally, we explored the real-world applications of logarithms in fields such as acoustics, seismology, chemistry, computer science, and finance, illustrating the practical significance of this mathematical concept.

Mastering logarithmic equations is a valuable skill that will benefit you in various academic and professional pursuits. We encourage you to practice solving similar equations to reinforce your understanding and build confidence in your problem-solving abilities. With a solid grasp of the properties of logarithms and a careful, methodical approach, you can successfully tackle a wide range of logarithmic problems.