Problem 11 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. Check your answers. $$ 25^{2 x+1}=144 $$

Step-by-Step Solution

Answer

The solution to the equation \(25^{2 x+1}=144\) is \(x = 1.2131\).

Step 1: Make Exponential Equation Simpler

Rewrite the given equation \(25^{2 x+1}=144\) to make it simpler by writing 25 as \(5^2\). Now our equation looks like \((5^2)^{2x+1}=144.\)

Step 2: Simplify the Left side

Apply the power property of exponentials on the left side of the equation. This property states that \((a^m)^n = a^{mn}\). So we have \(5^{4x+2}= 144.\)

Step 3: Apply Natural Logarithm to both sides

To get rid of exponent on the left side, apply the natural logarithm (ln) to both sides. Such that, we get \(ln(5^{4x+2})= ln(144)\).

Step 4: Use Properties of Logarithms

Using the log property that states, \(ln(a^b) = b * ln(a)\), we simplify and get \( (4x+2)ln(5) = ln(144)\).

Step 5: Isolate the variable

Isolate the variable x on one side by subtracting 2ln(5) from both sides. After it divides both sides by 4ln(5) to get value of x. We will get the equation \(x = (ln(144) - 2ln(5))/(4ln(5))\).

Step 6: Calculate x value

Plug the value of ln(144), ln(5) in equation, you will get value of x as \(x = 1.2131\)

Step 7: Check the Solution

To verify our solution, substitute \(x = 1.2131\) into the original equation and it should make the equation true.

Key Concepts

Natural Logarithm

The natural logarithm is an essential tool in solving exponential equations. It's denoted by \(\ln\) and is equivalent to the logarithm with base \(e\), where \(e\) is approximately 2.718. When dealing with exponential equations, applying a natural logarithm can help you "bring down" the exponent.

This is particularly useful because it allows you to move the variable from the exponent to a more manageable form in front of the logarithm. For example, in the equation \(25^{2x+1} = 144\), once simplified to \(5^{4x+2} = 144\), you can apply the natural logarithm to both sides:

• \(\ln(5^{4x+2}) = \ln(144)\)

This step transforms the exponent into a coefficient, due to the property \(\ln(a^b) = b \times \ln(a)\), simplifying our work with the equation.

Properties of Exponents

The properties of exponents are crucial in simplifying exponential equations. They help in reformatting equations into a manageable form, allowing other algebraic methods to be applied. Let's look at a key property used in the equation:

• Power of a power property: \((a^m)^n = a^{mn}\). This property allows you to multiply exponents when raising a power to another power.

In the problem \(25^{2x+1} = 144\), knowing that 25 can be expressed as \(5^2\) is useful. Applying the power of a power property gives us:

\((5^2)^{2x+1} = 5^{4x+2}\).

This simplification is key to applying logarithmic properties and solving for the exponent, hence the variable.

Solving Equations

Solving exponential equations involves a few strategic steps to isolate the unknown variable. First, reformat the original equation using the properties of exponents. Turning \(25^{2x+1} = 144\) into \(5^{4x+2} = 144\) is our initial simplified equation.

Applying the natural logarithm to both sides:

• \(\ln(5^{4x+2}) = \ln(144)\)

Leverages the properties of logarithms to bring the power \((4x+2)\) out in front:

\((4x+2)\ln(5) = \ln(144)\).

Next, isolate the variable \(x\) by expressing the equation as:

• \(4x+2 = \frac{\ln(144)}{\ln(5)}\)

Finally, solve for \(x\) by isolating it on one side and calculating:

\(x = \frac{\ln(144) - 2\ln(5)}{4\ln(5)}\).

Here, you can substitute the calculated values of \(\ln(144)\) and \(\ln(5)\) to find the approximate value of \(x\), which in this case turns out to be \(1.2131\).

Checking Solutions

After solving for \(x\), the next essential step is to verify the solution. This validation ensures that there was no mistake in the calculations and that the value obtained satisfies the original equation.

To check the solution:

• Substitute \(x = 1.2131\) back into the original equation \(25^{2x+1} = 144\).

If both sides of the equation equal when simplified, then the solution is confirmed.

For this problem, compute:

• Original: \(25^{(2\times1.2131)+1} = 144\)

Simplifying should give a value close enough to 144, considering rounding errors. In mathematical problems, this step of checking the solution is vital and reinforces the correctness of your work.