Solving $4^{x-5} = 6$ For X Using The Change Of Base Formula

Introduction

In this article, we will delve into solving the exponential equation $4^{x-5} = 6$ for $x$. Exponential equations often require the use of logarithms to isolate the variable. We'll explore how to use the change of base formula, a powerful tool for manipulating logarithms, to find the solution. The change of base formula, $\log_b y = \frac{\log y}{\log b}$, allows us to convert logarithms from one base to another, which is particularly useful when dealing with calculators that typically only compute logarithms base 10 (common logarithm) or base $e$ (natural logarithm).

Understanding Exponential Equations and Logarithms

Before we dive into the solution, let's briefly recap exponential equations and logarithms. An exponential equation is an equation where the variable appears in the exponent. For example, $a^x = b$ is an exponential equation, where $a$ is the base, $x$ is the exponent, and $b$ is the result. Logarithms are the inverse operation to exponentiation. The logarithm of a number $b$ to the base $a$ is the exponent to which $a$ must be raised to produce $b$. Mathematically, if $a^x = b$, then $\log_a b = x$. Understanding this relationship is crucial for solving exponential equations. The change of base formula allows us to express logarithms in different bases, making it easier to compute them using calculators or to simplify expressions. This formula is especially helpful when the base of the logarithm is not readily available on a calculator, as we can convert it to a more convenient base, such as base 10 or base $e$.

Steps to Solve Exponential Equations

Solving exponential equations generally involves the following steps:

1. Isolate the exponential term: Make sure the exponential term (the term with the variable in the exponent) is isolated on one side of the equation.
2. Take the logarithm of both sides: Apply a logarithm to both sides of the equation. The base of the logarithm can be any convenient base, but often base 10 or base $e$ is used.
3. Use logarithm properties to simplify: Use properties of logarithms, such as the power rule ($\log_a b^c = c \log_a b$), to simplify the equation and bring the variable down from the exponent.
4. Solve for the variable: Perform algebraic manipulations to isolate and solve for the variable.
5. Check the solution: Substitute the solution back into the original equation to verify that it is correct.

Applying the Change of Base Formula

The change of base formula is a cornerstone in manipulating and evaluating logarithms. This formula, expressed as $\log_b a = \frac{\log_c a}{\log_c b}$, allows us to switch from one logarithmic base ($b$) to another ($c$). This is particularly useful because calculators typically only compute logarithms in base 10 (denoted as $\log$ or $\log_{10}$) and base $e$ (the natural logarithm, denoted as $\ln$ or $\log_e$). The ability to change bases means we can evaluate logarithms with any base using a calculator. For instance, if we need to find $\log_5 16$, we can rewrite it as $\frac{\log 16}{\log 5}$ or $\frac{\ln 16}{\ln 5}$, both of which can be directly computed using a calculator. The change of base formula stems from the fundamental relationship between logarithms and exponents. By understanding and applying this formula, we gain significant flexibility in solving equations and simplifying expressions involving logarithms.

Solving $4^{x-5} = 6$ Step-by-Step

Now, let's solve the equation $4^{x-5} = 6$ using the change of base formula. We'll break down the solution into clear, manageable steps.

1. Isolate the exponential term: In this case, the exponential term $4^{x-5}$ is already isolated on the left side of the equation.

2. Take the logarithm of both sides: Apply the logarithm to both sides. We can use any base, but let's use the common logarithm (base 10) for simplicity:

   $$\log(4^{x-5}) = \log(6)$$

3. Use logarithm properties to simplify: Apply the power rule of logarithms, which states that $\log_a b^c = c \log_a b$:

   $$(x-5)\log(4) = \log(6)$$

4. Solve for the variable: Divide both sides by $\log(4)$:

   $$x-5 = \frac{\log(6)}{\log(4)}$$

   Now, isolate $x$ by adding 5 to both sides:

   $$x = \frac{\log(6)}{\log(4)} + 5$$

5. Calculate the value: Use a calculator to find the values of $\log(6)$ and $\log(4)$:

   $$\log(6) \approx 0.7782$$

   $$\log(4) \approx 0.6021$$

   Substitute these values into the equation:

   $$x \approx \frac{0.7782}{0.6021} + 5$$

   $$x \approx 1.2925 + 5$$

   $$x \approx 6.2925$$

Therefore, the solution to the equation $4^{x-5} = 6$ is approximately $x \approx 6.2925$.

Alternative Approach Using Natural Logarithms

We can also solve the same equation using natural logarithms (base $e$). This approach is equally valid and demonstrates the flexibility of using different logarithmic bases.

1. Take the natural logarithm of both sides:

   $$\ln(4^{x-5}) = \ln(6)$$

2. Use logarithm properties to simplify: Apply the power rule of logarithms:

   $$(x-5)\ln(4) = \ln(6)$$

3. Solve for the variable: Divide both sides by $\ln(4)$:

   $$x-5 = \frac{\ln(6)}{\ln(4)}$$

   Isolate $x$ by adding 5 to both sides:

   $$x = \frac{\ln(6)}{\ln(4)} + 5$$

4. Calculate the value: Use a calculator to find the values of $\ln(6)$ and $\ln(4)$:

   $$\ln(6) \approx 1.7918$$

   $$\ln(4) \approx 1.3863$$

   Substitute these values into the equation:

   $$x \approx \frac{1.7918}{1.3863} + 5$$

   $$x \approx 1.2925 + 5$$

   $$x \approx 6.2925$$

As we can see, the solution obtained using natural logarithms is the same as the solution obtained using common logarithms, which further illustrates the power and consistency of the change of base formula and logarithmic properties.

Verification of the Solution

To ensure the accuracy of our solution, it's essential to verify it by substituting the calculated value of $x$ back into the original equation. This step helps to catch any potential errors made during the solving process. Substituting $x \approx 6.2925$ into the original equation $4^{x-5} = 6$, we get:

$$4^{6.2925-5} = 4^{1.2925}$$

Using a calculator to evaluate $4^{1.2925}$, we find that it is approximately equal to 6:

$$4^{1.2925} \approx 6$$

This confirms that our solution $x \approx 6.2925$ is correct. Verification is a crucial step in solving any mathematical problem, as it provides confidence in the accuracy of the result. By substituting the solution back into the original equation, we ensure that the equation holds true and that no algebraic or computational errors were made during the solving process. This practice reinforces the understanding of the problem and the solution, making it an integral part of effective problem-solving.

Common Mistakes and How to Avoid Them

When solving exponential and logarithmic equations, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and arrive at the correct solution. One frequent error is incorrectly applying the properties of logarithms. For instance, students might mistakenly think that $\log(a + b)$ is equal to $\log(a) + \log(b)$, which is incorrect. The correct property is $\log(ab) = \log(a) + \log(b)$. Another common mistake is mishandling the change of base formula. It's crucial to remember that $\log_b a = \frac{\log_c a}{\log_c b}$, not $\frac{\log_c b}{\log_c a}$. When solving equations, it's vital to perform the same operation on both sides to maintain equality. Forgetting this principle can lead to incorrect solutions. Additionally, not checking the solution can result in accepting extraneous roots, especially in logarithmic equations where the domain is restricted to positive numbers. To avoid these mistakes, always double-check the properties of logarithms being used, pay close attention to the correct application of the change of base formula, ensure operations are performed consistently on both sides of the equation, and diligently verify the solution by substituting it back into the original equation. These practices will significantly enhance accuracy and confidence in solving exponential and logarithmic equations.

Conclusion

In summary, we have successfully solved the equation $4^{x-5} = 6$ for $x$ using the change of base formula. We demonstrated the step-by-step process, which includes isolating the exponential term, taking the logarithm of both sides, applying logarithmic properties, solving for the variable, and verifying the solution. We also explored an alternative approach using natural logarithms and highlighted the importance of the change of base formula in manipulating logarithms. Understanding and applying the concepts discussed in this article will equip you with the skills necessary to tackle a wide range of exponential and logarithmic equations. The ability to solve these types of equations is fundamental in many areas of mathematics and its applications, such as in calculus, physics, engineering, and economics. By mastering these techniques, you can confidently approach more complex problems and gain a deeper appreciation for the power and versatility of logarithms and exponential functions.