Solving The Exponential Equation 4 = 7^(2x) A Step By Step Guide

Solving exponential equations often requires the use of logarithms, especially when the bases cannot be easily made the same. In this article, we will walk through the process of solving the equation $4 = 7^{2x}$ and expressing the solution in its simplest form using integers, fractions, and common logarithms. Understanding exponential equations and how to solve them is a critical skill in mathematics, with applications ranging from finance to physics. Exponential equations model various real-world phenomena, including population growth, radioactive decay, and compound interest. Therefore, mastering the techniques to solve these equations is essential for anyone studying mathematics, science, or engineering.

Understanding Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. The general form of an exponential equation is $a = b^{cx}$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we want to solve for. To solve such equations, we often use logarithms. A logarithm is the inverse operation to exponentiation. The logarithm of a number to a given base is the exponent to which we must raise the base to produce that number. Mathematically, if $b^y = x$, then $\log_b(x) = y$. Common logarithms, which use base 10, are particularly useful because they align with our decimal number system. Natural logarithms, which use base $e$ (Euler's number, approximately 2.71828), are also widely used, especially in calculus and more advanced mathematical contexts. However, for this particular problem, using common logarithms will simplify the process and allow us to express the solution in a straightforward manner.

When approaching an exponential equation, it’s crucial to identify the base and the exponent. In the equation $4 = 7^{2x}$, the base is 7, and the exponent is $2x$. Our goal is to isolate the variable $x$. To do this, we will take the logarithm of both sides of the equation. By applying logarithmic properties, we can bring the exponent down and solve for $x$. This method is a standard technique for solving exponential equations and is applicable to a wide range of problems. Understanding and applying this technique effectively will not only help in solving the given equation but will also build a strong foundation for tackling more complex problems in algebra and calculus. Remember, the key to solving exponential equations lies in the strategic use of logarithms to simplify the equation and isolate the variable of interest.

Steps to Solve 4 = 7^(2x)

Let's break down the solution step-by-step. First, we have the equation $4 = 7^{2x}$. To solve for $x$, we need to isolate the variable, which is currently in the exponent. The most effective way to do this is by taking the logarithm of both sides of the equation. Taking the logarithm helps us bring the exponent down as a multiplier, which simplifies the equation and allows us to solve for $x$. This is a fundamental technique in solving exponential equations, and it leverages the inverse relationship between exponential functions and logarithmic functions. By applying logarithms, we can transform the equation into a more manageable form where $x$ can be easily isolated.

1. Take the logarithm of both sides:

To begin, we apply the common logarithm (base 10) to both sides of the equation. This gives us $\log_{10}(4) = \log_{10}(7^{2x})$. The choice of using the common logarithm is primarily for convenience, as it aligns with the decimal system and is readily available on most calculators. However, any logarithm base could be used, such as the natural logarithm (base $e$). The key is to apply the same logarithmic function to both sides to maintain the equality. This step is crucial because it sets the stage for using the power rule of logarithms, which will allow us to bring the exponent $2x$ down as a coefficient.

2. Apply the power rule of logarithms:

The power rule of logarithms states that $\log_b(a^c) = c \log_b(a)$. Applying this rule to the right side of our equation, we get $\log_{10}(4) = 2x \log_{10}(7)$. This step is critical because it transforms the exponential term into a product, making it possible to isolate $x$. The power rule of logarithms is a fundamental tool in solving exponential equations and is derived from the basic properties of exponents and logarithms. By bringing the exponent down as a multiplier, we've effectively linearized the equation, making it easier to solve. This application of the power rule is a common technique used in various mathematical contexts involving exponential and logarithmic functions.

3. Isolate x:

To isolate $x$, we divide both sides of the equation by $2 \log_{10}(7)$. This gives us $x = \frac{\log_{10}(4)}{2 \log_{10}(7)}$. This step directly follows from the previous application of the power rule and is a straightforward algebraic manipulation. By dividing both sides by the coefficient of $x$, we successfully isolate the variable, which is our primary goal. This process is a standard algebraic technique used to solve equations, and it is essential for determining the value of $x$ in this context.

Expressing the Solution in Simplest Form

Now that we have $x = \frac{\log_{10}(4)}{2 \log_{10}(7)}$, we want to express this in the simplest form. Simplification often involves using logarithmic properties to combine or reduce terms. To further simplify the expression, we can use the property of logarithms that allows us to rewrite constants within the logarithmic function. Specifically, we can use the logarithm property $\log(a^b) = b \log(a)$ in reverse to try and combine or simplify terms. This property is particularly useful when dealing with ratios of logarithms, as it can sometimes lead to more concise expressions. In this case, we can attempt to apply this property to the numerator and denominator to see if we can achieve further simplification.

1. Simplify the logarithm in the numerator:

We can rewrite $\log_{10}(4)$ as $\log_{10}(2^2)$. Applying the power rule, this becomes $2 \log_{10}(2)$. So our equation now looks like $x = \frac{2 \log_{10}(2)}{2 \log_{10}(7)}$. This simplification is a common technique used to reduce the complexity of logarithmic expressions. By recognizing that 4 is a power of 2, we can apply the power rule of logarithms to transform $\log_{10}(4)$ into a more manageable form. This step not only simplifies the equation but also provides an opportunity to cancel out common factors, leading to a more concise solution. Simplification at this stage is crucial for ensuring that the final answer is expressed in its most basic and understandable form.

2. Cancel common factors:

We can see that there is a 2 in both the numerator and the denominator, so we can cancel these out. This simplifies the expression to $x = \frac{\log_{10}(2)}{\log_{10}(7)}$. This step is a straightforward algebraic simplification that involves dividing both the numerator and denominator by their common factor, which is 2 in this case. Cancelling common factors is a fundamental technique in simplifying fractions and expressions, and it ensures that the final answer is in its most reduced form. By removing the common factor, we obtain a more concise and elegant expression for $x$, which is easier to interpret and use in further calculations. This simplification step is crucial for presenting the solution in its simplest form and for ensuring clarity in mathematical communication.

Therefore, the simplest form of the solution is:

$x = \frac{\log_{10}(2)}{\log_{10}(7)}$

Final Answer

The solution to the equation $4 = 7^{2x}$, expressed in its simplest form using common logarithms, is $x = \frac{\log_{10}(2)}{\log_{10}(7)}$. This answer is exact and does not require further approximation unless a decimal value is specifically requested. Expressing solutions in terms of logarithms is a common practice in mathematics, particularly when dealing with exponential equations that do not have simple integer or fractional solutions. The use of logarithms allows us to provide an accurate and concise representation of the solution. To reiterate, the final answer is $x = \frac{\log_{10}(2)}{\log_{10}(7)}$, which represents the exact value of $x$ that satisfies the original equation.

# Solve the exponential equation 4 = 7^(2x) and express the solution in simplest form

4 = 7^(2x) x = [ ]

What are the steps to solve the exponential equation 4 = 7^(2x) and express the solution in the simplest form using integers, fractions, and common logarithms?