Problem 35 - Intermediate Algebra

Question

Solve each equation. Give the exact solution and an approximation to four decimal places. $$ 5^{x-3}=3^{2 x} $$

Step-by-Step Solution

Answer

Exact: \(x = \frac{3 \ln(5)}{\ln(5) - 2\ln(3)}\), Approximate: \(x \approx 4.1231\).

Step 1: Take the logarithm of both sides

To solve the equation \(5^{x-3}=3^{2x}\), start by taking the natural logarithm of both sides to bring the exponents down. This gives us an equation that we can work with more easily: \(\ln(5^{x-3}) = \ln(3^{2x})\).

Step 2: Apply the power rule for logarithms

Use the power rule \(\ln(a^b) = b\ln(a)\) to each side of the equation. This results in \((x-3)\ln(5) = 2x\ln(3)\).

Step 3: Distribute and simplify the equation

Distribute the \(\ln(5)\) in the left side of the equation: \(x \ln(5) - 3\ln(5) = 2x \ln(3)\).

Step 4: Gather like terms

Rearrange the equation to gather all terms involving \(x\) on one side. This looks like \(x \ln(5) - 2x \ln(3) = 3 \ln(5)\).

Step 5: Factor out \(x\)

Factor \(x\) out of the left side of the equation: \(x (\ln(5) - 2\ln(3)) = 3 \ln(5)\).

Step 6: Solve for x

Divide both sides by \((\ln(5) - 2\ln(3))\) to isolate \(x\): \(x = \frac{3 \ln(5)}{\ln(5) - 2\ln(3)}\).

Step 7: Calculate the exact and approximate values

Evaluate the expression for \(x\) to find the exact value and then use a calculator to find an approximation: \(x \approx \frac{3 \ln(5)}{\ln(5) - 2\ln(3)}\). Approximating gives \(x \approx 4.1231\).

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical function commonly denoted as \( \ln(x) \). It is the logarithm to the base \( e \), where \( e \approx 2.71828 \) and is known as Euler's number. Natural logarithms are significant because they have properties that simplify calculations involving exponentials. This is especially useful in solving exponential equations, like the one we have here: \( 5^{x-3} = 3^{2x} \).

• Natural logarithms help us "bring down" the exponent which can simplify the equation.
• They are widely used in calculus, growth models, and even complex mathematical theories.

In our problem, using \( \ln(5^{x-3}) = \ln(3^{2x}) \), allows equilibration of the equation in a simpler linear form, facilitating resolution.

Power Rule for Logarithms

The power rule for logarithms is a foundational tool in algebra. It states that for any positive number \( a \) and real number \( b \), the log of \( a \) raised to the power of \( b \) can be expressed as \( b \) times the log of \( a \): \( \ln(a^b) = b \ln(a) \).

• This rule is vital in transforming equations with variables in the exponent to linear equations.
• It simplifies separating the power from the base, enabling easier manipulation of the equation.

In our equation, applying the power rule changes \( \ln(5^{x-3}) \) to \( (x-3)\ln(5) \) and \( \ln(3^{2x}) \) to \( 2x\ln(3) \). This converts an otherwise complex equation into a form we can solve using algebraic methods.

Exact Solutions

An exact solution is one that is precise and accurate without any approximation. When dealing with equations like \( 5^{x-3} = 3^{2x} \), it's essential to first derive the exact solution using algebraic methods before moving to find any approximations.

• Create an equation in terms of \( x \) by isolating \( x \).
• Use algebraic manipulation to simplify and solve for \( x \).

In our context, we isolate \( x \) by factoring and simplifying the linear equation derived from taking logarithms. This gives us the exact expression \( x = \frac{3 \ln(5)}{\ln(5) - 2\ln(3)} \), where \( x \) is precisely expressed in terms of naturally occurring numbers.

Approximation of Solutions

After determining the exact solution, it's often necessary to approximate the answer to make it more digestible. Approximations are numerical estimations of exact formulas, useful when exact values are either too complex or unnecessary for practical purposes.

• Use a calculator to find approximate decimal values.
• Round to a specific precision, like four decimal places.

In our case, we calculated \( x = \frac{3 \ln(5)}{\ln(5) - 2\ln(3)} \) and found that \( x \approx 4.1231 \) when rounded to four decimal places. Approximations are crucial in engineering, science, and finance where precise estimations are based on numerous variables.