Problem 41 - College Algebra Essentials

Question

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5^{2 x+3}=3^{x-1}$$

Step-by-Step Solution

Answer

\( x = \frac{-ln(3)-3ln(5)}{2ln(5)-ln(3)} \)

Step 1: Apply Logarithm to Both Sides of the Equation

First, apply the natural logarithm (ln) to both sides of the equation, this helps to break down the equation. Using properties of logarithms, \( ln(5^{2x+3})=ln(3^{x-1})\) becomes \( (2x+3)ln(5) = (x-1)ln(3)\)

Step 2: Simplify the Equation

Now, redistribute the logarithms to obtain a simpler equation: \( 2xln(5)+3ln(5)=xln(3)-ln(3)\)

Step 3: Collect Like Terms

Move the term involving x in one side of the equation to group the ones with the same variable together and the constant terms on the other side: \( 2xln(5)-xln(3)=-ln(3)-3ln(5)\)

Step 4: Factor Out the x

Then factor out the x term, we get: \( x(2ln(5)-ln(3))=-ln(3)-3ln(5) \)

Step 5: Solve for x

Lastly, we can now solve for x by dividing by \(2ln(5)-ln(3)\) on both sides, so we get: \( x = \frac{-ln(3)-3ln(5)}{2ln(5)-ln(3)} \)

Step 6: Decimal Approximation

To obtain a decimal approximation correct to two decimal places, use a calculator to compute the value of x