Problem 11 - Precalculus with Calculus Previews

Question

Solve the given exponential equation. $$ 5-10^{2 x}=0 $$

Step-by-Step Solution

Answer

The solution is approximately \(x \approx 0.349485\).

Step 1: Isolate the Exponential Term

The given equation is \(5 - 10^{2x} = 0\). Our goal is to isolate the exponential part \(10^{2x}\). To do this, subtract 5 from both sides of the equation:\[ 10^{2x} = 5 \]

Step 2: Apply the Logarithm

To solve for \(x\), we need to eliminate the exponential part by taking the logarithm of both sides. It is easiest to use the base-10 logarithm because our base is 10:\[ \log_{10}(10^{2x}) = \log_{10}(5) \]

Step 3: Simplify the Logarithm Expression

By the properties of logarithms, particularly that \(\log_{b}(b^c) = c\), the left side simplifies:\[ 2x = \log_{10}(5) \]

Step 4: Solve for x

To isolate \(x\), divide both sides by 2:\[ x = \frac{\log_{10}(5)}{2} \]

Step 5: Calculate the Numerical Value

Use a calculator to find the approximate value of \(\log_{10}(5)\), which is approximately 0.69897. Then divide by 2:\[ x \approx \frac{0.69897}{2} = 0.349485 \]

Key Concepts

Logarithms

Logarithms are a special mathematical function that help us deal with equations where the variable is in the exponent. Essentially, logarithms are the inverse operations of exponentiations. For instance, if we have an equation of the form \( b^c = a \), taking the logarithm of both sides can isolate the exponent, \( c \). This means \( c = \log_b(a) \), where \( b \) is the base of the logarithm and \( a \) is the value we are taking the logarithm of. Logarithms have multiple properties that make them very useful:

• \( \log_b(b^c) = c \) - This property, called the power rule, simplifies equations by bringing the exponent down.
• \( \log_b(xy) = \log_b(x) + \log_b(y) \) - This is the product rule, which turns multiplication inside the log into addition outside it.
• \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \) - The quotient rule helps in breaking down divisions inside the logarithm.

In our example, taking the logarithm of \( 10^{2x} \) with base 10 simplifies to \( 2x \). This property can greatly simplify solving exponential equations.

Isolate Exponential Term

To effectively solve an exponential equation, one of the primary steps is to isolate the exponential term. In our exercise, we start with the equation \( 5 - 10^{2x} = 0 \). Here is how to isolate it step-by-step:

• First, observe that \(10^{2x}\) is the exponential term. It's the expression we need to get by itself on one side of the equation.
• Move other terms (in this case, the constant 5) to the opposite side of the equation. We do this by subtracting 5 from both sides, resulting in \(10^{2x} = 5\).

Isolating the exponential term is crucial because once it is alone, you can apply logarithms or other methods to solve for the variable.

Solve for Variable

Once the exponential term is isolated and the logarithm has been applied, it's time to solve for the variable. Our goal is to find the value of \( x \) in the equation derived from the original exercise. Here’s how to proceed:

• After taking the logarithm of both sides, you get an equation like \( 2x = \log_{10}(5) \).
• This equation tells us that the original exponential expression equals the logarithmic result on the right side.
• The variable \( x \) is currently multiplied by 2. To isolate \( x \), divide both sides by 2.
• Now, the equation becomes \( x = \frac{\log_{10}(5)}{2} \).

For the final numerical approximation, calculate the logarithm of 5 using a calculator, which is roughly 0.69897. Divide that by 2 to find \( x \approx 0.349485 \). This solution describes the value of \( x \) for the original exponential equation. Solving for \( x \) this way ensures a clear, precise outcome.