Problem 67 - College Algebra with Modeling and Visualization

Question

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$4\left(3^{x}\right)-3=13$$

Step-by-Step Solution

Answer

The solution is approximately \(x = 1.26\).

Step 1: Isolate the Exponential Expression

Start with the equation \(4(3^x) - 3 = 13\). Add 3 to both sides to get \(4(3^x) = 16\). Then divide by 4 to isolate the exponential term, giving \(3^x = 4\).

Step 2: Apply the Change of Base Formula

To find \(x\), rewrite the equation \(3^x = 4\) using logarithms. Applying the natural logarithm on both sides, we have \(\ln(3^x) = \ln(4)\). By the properties of logarithms, \(x \cdot \ln(3) = \ln(4)\).

Step 3: Solve for x

Divide both sides by \(\ln(3)\) to solve for \(x\): \(x = \frac{\ln(4)}{\ln(3)}\).

Step 4: Approximate the Answer

Use a calculator to find \(\ln(4) \approx 1.3863\) and \(\ln(3) \approx 1.0986\). Divide these values to approximate \(x\): \(x \approx \frac{1.3863}{1.0986} \approx 1.26\).

Key Concepts

Change of Base Formula

The change of base formula is a handy tool in mathematics, especially when dealing with logarithms. It allows you to convert a logarithm from one base to another. This is particularly useful when solving exponential equations, as it helps to simplify calculations. The formula is given by:

• \(\log_b a = \frac{\log_k a}{\log_k b}\)

Where \(b\) is the base you are converting from, \(a\) is the number you are taking the logarithm of, and \(k\) is the new base, often e (natural logarithm) or 10 (common logarithm).
To apply the formula, you essentially find the logarithms of your number and base using your new base and divide them. This simplifies complex exponential equations where the exponent is unknown, and precise computation is needed for solutions.

Logarithms

Logarithms are the inverse operations of exponential functions. In simpler terms, if you have an equation of the form \(b^x = a\), a logarithm helps you to solve for \(x\) by expressing it as \(x = \log_b a\). The base \(b\) in this context matches the base in the original exponential equation.
Logarithms have several important properties:

• The product property \(\log_b(mn) = \log_b m + \log_b n\).
• The quotient property \(\log_b(\frac{m}{n}) = \log_b m - \log_b n\).
• The power property \(\log_b(m^n) = n \cdot \log_b m\).

These properties are essential as they facilitate breaking down and simplifying equations, making them more manageable to solve. You often use the natural logarithm \(\ln\) (base \(e\)) when solutions require high precision, as with constants such as 2.71828.

Solving Equations

Solving exponential equations involves isolating the exponential expression first. This set-up pivots the equation into a solvable form, often utilizing logarithms for simplicity. Take the problem:

• \(4(3^x) - 3 = 13\)

To solve this, start by isolating \(3^x\) by adding 3 to each side, obtaining \(4(3^x) = 16\). Dividing each side by 4, it simplifies to \(3^x = 4\).
Here, the aim is to solve for \(x\) by applying logarithms to both sides, using the change of base formula if necessary. Thus, taking the natural logarithm on both sides, you reach:

• \(\ln(3^x) = \ln(4)\)
• \(x \cdot \ln(3) = \ln(4)\)

Finally, solve for \(x\) by dividing both sides by \(\ln(3)\). Use a calculator to find these values, thus achieving a precise approximation of \(x\). The final calculation yields approximately \(x \approx 1.26\), granting an efficient solution using logarithms.