Problem 46 - Algebra 2

Question

Solve each equation. Round to four decimal places. $$ 2^{2 x+3}=3^{3 x} $$

Step-by-Step Solution

Answer

The solution is approximately \(x \approx 1.0773\).

Step 1: Take the Logarithm of Both Sides

To simplify the exponential equation, apply the logarithm to both sides. \[\log(2^{2x+3}) = \log(3^{3x})\]

Step 2: Use Logarithmic Properties

Using the power rule of logarithms, \(\log(a^b) = b\log(a)\), rewrite the equation as: \[(2x+3)\log(2) = 3x \log(3)\]

Step 3: Distribute and Rearrange the Equation

Distribute the logarithm across the terms within the brackets and rearrange:\[2x \log(2) + 3 \log(2) = 3x \log(3)\]Subtract \(2x \log(2)\) from both sides:\[3 \log(2) = 3x \log(3) - 2x \log(2)\]

Step 4: Factor Out x

Factor out \(x\) from the right side of the equation:\[3 \log(2) = x (3 \log(3) - 2 \log(2))\]

Step 5: Solve for x

Isolate \(x\) by dividing both sides of the equation by \((3 \log(3) - 2 \log(2))\):\[x = \frac{3 \log(2)}{3 \log(3) - 2 \log(2)}\]Calculate this expression using a calculator and round to four decimal places.

Key Concepts

Logarithms

Logarithms help us solve equations involving exponential expressions. They allow us to transform exponential numbers in a way that can be more easily managed mathematically. A logarithm answers the question: "To what power must we raise a certain number (the base) to get another number?"
For example, if we have an equation involving powers of 2 or 3, using logarithms can simplify the expression. This can make solving the equation simpler.

Power Rule of Logarithms

The power rule of logarithms is a useful property when dealing with exponential equations. It states that \[ \log(a^b) = b \log(a) \]This rule is powerful because it allows us to bring exponents down, effectively linearizing the equation.
This means that instead of dealing with a complicated exponent, we can convert it into a multiplication problem that's easier to handle.

Solving Equations

Solving exponential equations often involves transforming the equation so it's easier to work with. By taking the logarithm of both sides of an equation, we simplify exponential terms. This, combined with the power rule of logarithms, allows for subsequent steps like factoring or isolating variables.

• First, apply logarithms to both sides of the equation to make use of properties like the power rule.
• Second, distribute and rearrange terms for clarity and simplicity.
• Finally, isolate the variable to solve for its value.

This method is systematic and can be applied to many exponential equations.

Rounding Decimal Places

Rounding is crucial when dealing with real-world applications where precise numbers may not be necessary or practical. After computing values, especially in logarithmic calculations, rounding to a specific number of decimal places, like four in this example, ensures consistency and simplicity.
To round a number to four decimal places:

• Look at the fifth decimal place. If it is 5 or greater, increase the fourth decimal place by one.
• If it is less than 5, keep the fourth decimal place as is.

Rounding helps to simplify results, making them easier to communicate and apply.