Problem 19 - Precalculus with Limits

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(4\left(3^{x}\right)=20\)

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 1.465\)

Step 1: Isolate the exponential term.

The first step is to isolate \(3^x\) on one side. It's done by dividing both sides of the equation by 4. This gives: \(3^x = \frac{20}{4}\)

Step 2: Simplify the equation

Reducing \(\frac{20}{4}\) gives us 5. So the equation now becomes: \(3^x = 5\)

Step 3: Apply Logarithm to both sides

To further isolate x, apply the natural logarithm (ln) to both sides of the equation: \[\ln(3^x) = \ln(5)\]

Step 4: Simplify using Logarithm identity

Using the identity \(\ln(a^b) = b\ln(a)\), the equation simplifies to: \(x\ln(3) = \ln(5)\)

Step 5: Solve for x

To isolate x, divide both sides of the equation by \(\ln(3)\). This gives: \(x = \frac{\ln(5)}{\ln(3)}\)

Step 6: Approximate the result

Using a calculator to find the natural logarithm of 5 and 3 and then divide the results, the result could be approximated to three decimal places: \(x = 1.465\)

Key Concepts

Logarithms

Logarithms are mathematical operations that are the inverse of exponentiation. If you have an equation such as \(a^b=c\), the logarithm helps you solve for \(b\). This is done by calculating \(b = \log_a(c)\).
This function essentially asks the question: "To what power must the base \(a\) be raised, to produce the number \(c\)?"

• It simplifies calculations involving large numbers by converting multiplication into addition.
• There are specific bases commonly used, such as base 10 (common logarithm) and base \(e\) (natural logarithm).

In our problem, once we've isolated the exponential term, we applied a logarithm to solve for \(x\). This reverses the exponential operation, bringing \(x\) out of the exponent so that we can solve for it directly.

Isolation of Variable

The isolation of a variable involves rearranging an equation so that the variable of interest is alone on one side of the equation.
This process is crucial in solving equations, as it helps you find the value of the unknown variable.
For the given equation \(4(3^x)=20\), the first step was to isolate the term \(3^x\).
This was achieved by dividing both sides by 4, resulting in the simplified equation \(3^x=5\).

• Isolating the variable involves using inverse operations like addition/subtraction or multiplication/division.
• This allows you to systematically solve for the unknown variable without altering the equation's balance.

Careful manipulation and understanding of the rules of algebra ensure accurate solutions during the isolation process.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), uses the mathematical constant \(e\) as its base. \(e\) is approximately 2.718 and is crucial in calculus and mathematical modeling.
The natural logarithm has unique properties that make it especially useful in math and science, often used when growth processes over time are involved, such as population growth or radioactive decay.
In our exercise, applying the natural logarithm to both sides of the equation \(3^x = 5\) was necessary for isolating \(x\).
By using the logarithmic identity \(\ln(a^b) = b\ln(a)\), we simplified our equation to \(x\ln(3) = \ln(5)\).

• This allows the variable \(x\) to be solved simply by dividing \(\ln(5)\) by \(\ln(3)\).
• Natural logarithms help manage exponential expressions elegantly, thus streamlining the calculation process.

Understanding how to apply these properties simplifies many equations involving exponential relationships.