Problem 12 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(4\left(1+10^{5 x}\right)=9\)

Step-by-Step Solution

Answer

The solution is approximately \(x = 0.0175\).

Step 1: Simplify the Equation

The given equation is \(4(1+10^{5x})=9\). Start by dividing both sides of the equation by 4 to isolate the exponential expression. This gives us: \(1 + 10^{5x} = \frac{9}{4}\).

Step 2: Isolate the Exponential Term

Subtract 1 from both sides of the equation to further isolate the exponential term. \(10^{5x} = \frac{9}{4} - 1\). Calculate \(\frac{9}{4} - 1 = \frac{5}{4}\). So, \(10^{5x} = \frac{5}{4}\).

Step 3: Apply the Logarithm

To solve for \(x\), take the common logarithm (base 10) on both sides: \(\log(10^{5x}) = \log\left(\frac{5}{4}\right)\).

Step 4: Simplify Using Logarithmic Identity

Apply the power rule of logarithms, \(\log(a^b) = b\cdot\log(a)\), to simplify as follows: \(5x \cdot \log(10) = \log\left(\frac{5}{4}\right)\). Since \(\log(10) = 1\), this simplifies to \(5x = \log\left(\frac{5}{4}\right)\).

Step 5: Solve for x

Divide both sides by 5 to solve for \(x\): \(x = \frac{1}{5} \cdot \log\left(\frac{5}{4}\right)\).

Step 6: Calculate Numerical Solution

Use a calculator to find \(\log\left(\frac{5}{4}\right)\) and then divide by 5. This results in \(x \approx 0.0175\) when rounded to four decimal places.

Key Concepts

Understanding Logarithms

Logarithms are mathematical tools used to solve equations involving exponential expressions. They work by transforming the equation into a form that's easier to manage. Logarithms are essentially the inverse of exponentiation. If you have a number like \(10^x = y\), the logarithmic form is \(x = \log_{10}(y)\). Here, the base of the logarithm is 10, which is the most common base used in calculations.
In our original problem, we applied the logarithm to both sides after isolating the exponential term, \(10^{5x}\). By taking the logarithm, you simplify the expression in a way that lets you use regular arithmetic to solve for the variable, in this case, \(x\). Logarithms help by breaking down the exponentiation, allowing us to bring down exponents for easier manipulation.

Power Rule of Logarithms

The power rule of logarithms is a crucial property that helps simplify expressions where exponents are involved. It states that \(\log(a^b) = b \cdot \log(a)\). This rule is derived from the basic properties of logarithms and allows you to move the exponent of a logarithmic term to the front, turning it into a multiplication.
In our problem, once we took the logarithm of both sides, \(\log(10^{5x})\) became \(5x \cdot \log(10)\). Using the power rule, the exponent \(5x\) is "brought down," simplifying the logarithmic expression. This step transforms a complex problem back into a simpler linear form, which is much more straightforward to solve.
The power rule is especially helpful when dealing with exponential equations, as it bridges the gap between exponential and logarithmic forms.

Solving Exponential Equations

Exponential equations involve expressions where variables are in the exponent, and solving them often requires several strategic steps:

• Simplify the Equation: Begin by isolating the exponential term if possible. Simplify the equation by dividing or subtracting to get the exponential expression with a coefficient of one.
• Apply Logarithms: Use logarithms to transform the equation into one where the variable can be accessed. Taking the logarithm of both sides is a common technique.
• Simplify Using Rules: Use properties of logarithms, like the power rule, to simplify the expression, making it solvable through basic algebra.
• Solve for the Variable: Finally, use normal algebraic methods, like division or multiplication, to find the value of the variable.

In the given problem, each of these steps was crucial. We simplified by dividing by 4, isolated \(10^{5x}\), used logarithms to transform the equation, and then applied the power rule before solving for \(x\). These steps enabled the calculation to move smoothly from a complex form to a final solution.