Problem 39 - Precalculus

Question

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$\frac{4}{10^{2 x}-7}=2$$

Step-by-Step Solution

Answer

The solution is approximately \( x \approx 0.477 \).

Step 1: Isolate the Base Expression

First, we need to isolate the expression that contains the base. The given equation is \( \frac{4}{10^{2x} - 7} = 2 \). To clear the fraction, multiply both sides of the equation by \( 10^{2x} - 7 \): \[ 4 = 2(10^{2x} - 7) \]

Step 2: Simplify the Equation

Next, distribute the 2 on the right side of the equation:\[ 4 = 2 \cdot 10^{2x} - 14 \]Add 14 to both sides to isolate the term with the exponent:\[ 18 = 2 \cdot 10^{2x} \]

Step 3: Solve for the Exponential Term

Divide both sides by 2 to solve for \( 10^{2x} \):\[ 10^{2x} = 9 \]

Step 4: Apply Logarithms

Take the logarithm of both sides to solve for \( x \). We use the property of logarithms that \( \log(a^b) = b \log(a) \):\[ \log(10^{2x}) = \log(9) \]\[ 2x \cdot \log(10) = \log(9) \] Since \( \log(10) = 1 \), the equation simplifies to:\[ 2x = \log(9) \]

Step 5: Calculate the Value of x

To find \( x \), divide both sides by 2:\[ x = \frac{\log(9)}{2} \]Using a calculator, we find \( \log(9) \approx 0.954 \), hence:\[ x \approx \frac{0.954}{2} \approx 0.477 \]

Key Concepts

Logarithms

Logarithms are powerful tools in mathematics, especially when dealing with exponential equations like our exercise example. They allow us to convert multiplicative relationships into additive ones, simplifying the process of solving exponential equations. A logarithm can be thought of as the inverse operation of an exponent.

In the equation given, we reached a stage where it was necessary to solve for an unknown exponent. To achieve this, we applied the property:

• If you have an equation of the form \( a^b = c \), then \( b = \log_a(c) \).

This property is essential when isolating exponential terms. In our case, we used the common logarithm, which employs base 10, simplifying since \( \log_{10} \) of its base is 1. This made it straightforward to solve the equation for the unknown variable by applying the relationship \( \log(a^b) = b \log(a) \).

Understanding logarithms can be vital in various fields, from science to finance, where exponential functions frequently model real-world situations.

Mathematical Solutions

The solution process for exponential equations involves multiple steps to simplify and solve for the unknown variable. In the provided example, it’s crucial to follow a structured approach:

• **Fraction Clearing:** It's essential to eliminate any fractions by multiplying through by the denominator. This step isolates the variable expression, making it more accessible to solve.
• **Expression Simplification:** Distribute any coefficients and combine like terms, setting the equation up to isolate the variable.
• **Solving for the Exponent:** Once you reach an exponential form, isolate it completely by removing any coefficients associated with the base expression.
• **Applying Logarithms:** Take logarithms to simplify the power equation, and solve for the variable using properties of logarithms.

By following these steps, one ensures that mistakes are minimized, and the solution process is logically structured. Each stage builds upon the last, bringing clarity to the path of reaching the final answer.

Equation Solving Techniques

Solving exponential equations requires a straightforward approach that combines algebraic techniques and the use of logarithms. When tackling these equations, there are several strategies to keep in mind:

• **Isolate the Base:** Start by isolating the base term of the exponential expression, as seen in our method that rearranged a fraction to simplify solving for the exponent.
• **Clear Complicated Expressions:** Remove any complex expressions by using algebraic techniques such as distributing, multiplying, or adding to both sides to ensure the variable stands alone.
• **Use Logarithms Efficiently:** Apply logarithms thoughtfully to transform the equation from an exponential form to a linear equation. This makes it easier to solve for the unknown variable.
• **Rounding Solutions:** Finally, round your solutions accurately to the desired decimal places, ensuring your solution meets any problem specifications.

These techniques don't only apply to the current problem but also to a wide range of mathematical challenges, particularly those involving exponential growth or decay models. By practicing these methods, solving complex equations becomes a more manageable task.