Rewriting Logarithmic Equations To Exponential Form A Comprehensive Guide

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This article delves into the process of rewriting logarithmic equations in their equivalent exponential forms. We will specifically address the equation $\log _4(x+6)=2$ and explore the correct method to express it using exponents. Understanding the relationship between logarithms and exponents is crucial for solving various mathematical problems, especially in algebra and calculus. This exploration aims to provide a comprehensive explanation that clarifies the underlying principles and steps involved in this transformation.

Let's consider the given logarithmic equation:

$\log _4(x+6)=2$

To rewrite this equation in exponential form, it’s essential to recall the fundamental relationship between logarithms and exponents. The logarithmic equation $\log _b(a) = c$ is equivalent to the exponential equation $b^c = a$. Here, $b$ is the base, $a$ is the argument, and $c$ is the exponent. Applying this understanding to our equation, we can identify the components:

• Base ($b$): 4
• Argument ($a$): $x+6$
• Exponent ($c$): 2

Using the exponential form $b^c = a$, we substitute the values to get:

$4^2 = x+6$

Thus, the original logarithmic equation $\log _4(x+6)=2$ can be rewritten in exponential form as $4^2 = x+6$. This transformation allows us to solve for $x$ by simplifying the exponential expression and isolating the variable. The key takeaway is recognizing the direct correspondence between logarithmic and exponential forms, enabling seamless conversion between the two.

Analyzing the Options

Now, let's examine the provided options to determine which one correctly represents the exponential form of the given logarithmic equation $\log _4(x+6)=2$.

Option A: $4^{\log _4(x+6)}=2^2$

This option starts by applying the base 4 as an exponent on both sides of the original equation. While the left side simplifies nicely due to the property $b^{\log _b(a)} = a$, the right side presents a potential misinterpretation. Let's break it down:

Starting with the original equation:

$\log _4(x+6)=2$

Raise 4 to the power of both sides:

$4^{\log _4(x+6)}=4^2$

Using the inverse property of logarithms, the left side simplifies to:

$x+6$

So the equation becomes:

$x+6 = 4^2$

$4^2$ equals 16, so:

$x+6 = 16$

Now, let's analyze option A:

$4^{\log _4(x+6)}=2^2$

The left side simplifies to $x+6$ as before:

$x+6 = 2^2$

$2^2$ equals 4, so:

$x+6 = 4$

This result is different from our correct transformation where $x+6 = 16$. Therefore, Option A is incorrect because it equates $x+6$ to 4, whereas the correct exponential form should equate $x+6$ to $4^2$ (which is 16).

In summary, while the initial step of raising 4 to the power of both sides is valid, the incorrect simplification on the right side leads to a different equation, making option A an inaccurate representation of the exponential form.

Option B: $4^{\log _4(x+6)}=4^4$

In this option, we again see the application of the base 4 as an exponent on the left side, which correctly utilizes the inverse relationship between logarithms and exponents. However, the right side, $4^4$, presents a significant deviation from the correct exponential form. Let's analyze step by step:

Starting from the original equation:

$\log _4(x+6) = 2$

Raise 4 to the power of both sides:

$4^{\log _4(x+6)} = 4^2$

As we established earlier, the left side simplifies to $x+6$ due to the inverse property of logarithms:

$x+6 = 4^2$

$4^2$ equals 16, so the correct transformation gives us:

$x+6 = 16$

Now, let's break down Option B:

$4^{\log _4(x+6)} = 4^4$

Simplifying the left side using the inverse property:

$x+6 = 4^4$

Here, $4^4$ equals 256. Thus, Option B implies:

$x+6 = 256$

This equation is fundamentally different from the correct exponential form $x+6 = 16$. Therefore, Option B is incorrect. The right side of the equation should have been $4^2$ to accurately represent the exponential form of the given logarithmic equation.

In conclusion, while the manipulation on the left side is correct, the erroneous use of $4^4$ on the right side makes Option B an incorrect representation of the exponential form of $\log _4(x+6) = 2$.

Option C: $\log _4(x+6)=\log _4 16$

This option presents a different approach compared to options A and B. It aims to rewrite the equation by expressing both sides in logarithmic form with the same base. To assess its correctness, we need to evaluate whether the right side, $\log _4 16$, is equivalent to the original right side of the equation, which is 2.

Let's start with the original equation:

$\log _4(x+6) = 2$

Now, we need to determine if 2 can be expressed as a logarithm with base 4. In other words, we need to find a value $y$ such that:

$\log _4 y = 2$

Using the exponential form, we rewrite this as:

$4^2 = y$

Therefore,

$y = 16$

So, the number 2 can indeed be expressed as $\log _4 16$. Thus, the original equation can be rewritten as:

$\log _4(x+6) = \log _4 16$

Comparing this with Option C:

$\log _4(x+6) = \log _4 16$

We can see that Option C correctly expresses the original equation with both sides in logarithmic form using the same base. This transformation is a valid method for solving logarithmic equations, as it allows us to equate the arguments of the logarithms:

$x+6 = 16$

This approach is consistent with the correct exponential transformation we derived earlier.

Therefore, Option C is correct. It accurately represents the given logarithmic equation by expressing both sides in terms of logarithms with the same base, which is a valid and useful technique in solving logarithmic equations.

Option D: $\log _4(x+6)=\log _2 16$

This option attempts to rewrite the equation in logarithmic form, similar to Option C, but it introduces a different base on the right side. Here, the left side remains the same, $\log _4(x+6)$, while the right side is expressed as $\log _2 16$. To determine the validity of this transformation, we need to check if $\log _2 16$ is equivalent to the original right side of the equation, which is 2.

Starting with the original equation:

$\log _4(x+6) = 2$

Now, let's evaluate $\log _2 16$. We need to find a value $z$ such that:

$2^z = 16$

We know that:

$2^4 = 16$

Therefore,

$\log _2 16 = 4$

Now, comparing this with the original equation:

$\log _4(x+6) = 2$

Option D states:

$\log _4(x+6) = \log _2 16$

Since we found that $\log _2 16 = 4$, Option D is essentially stating:

$\log _4(x+6) = 4$

This is different from the original equation where $\log _4(x+6) = 2$. Therefore, Option D is incorrect. It changes the fundamental relationship by equating $\log _4(x+6)$ to 4 instead of 2.

In summary, while expressing numbers in logarithmic form is a valid technique, the incorrect evaluation of $\log _2 16$ as being equivalent to the original right side (2) makes Option D an inaccurate representation of the given logarithmic equation.

Conclusion

After analyzing all the options, we can definitively conclude that Option C is the correct answer. Option C, $\log _4(x+6)=\log _4 16$, accurately represents the equation $\log _4(x+6)=2$ by expressing both sides in logarithmic form with the same base. This transformation allows for easy comparison and simplification, ultimately leading to the solution of the equation. Understanding the interplay between logarithmic and exponential forms is crucial in mathematics, and this problem serves as a great illustration of this concept.

Which equation correctly rewrites $\log _4(x+6)=2$ using exponents?

Rewriting Logarithmic Equations to Exponential Form A Comprehensive Guide