Question

solve the following exponential equation. express the solution in terms of natural logarithms or common logarithms. then, use a calculator to obtain a decimal approximation for the solution.
2^{5x + 4} = 3^{x - 3}
the solution set expressed in terms of logarithms is
(use a comma to separate answers as needed. simplify your answer. use integers or fractions for any numbers in the expression. use ln for natural logarithm and log for common logarithm.)

Explanation:

Step1: Take natural log on both sides

Take the natural logarithm of both sides of the equation \(2^{5x + 4}=3^{x - 3}\). By the property of logarithms \(\ln(a^b)=b\ln(a)\), we get \(\ln(2^{5x + 4})=\ln(3^{x - 3})\), which simplifies to \((5x + 4)\ln(2)=(x - 3)\ln(3)\).

Step2: Expand both sides

Expand the left - hand side and the right - hand side: \(5x\ln(2)+4\ln(2)=x\ln(3)-3\ln(3)\).

Step3: Collect like terms

Move all terms with \(x\) to one side and the constant terms to the other side. Subtract \(x\ln(3)\) from both sides and subtract \(4\ln(2)\) from both sides: \(5x\ln(2)-x\ln(3)=- 3\ln(3)-4\ln(2)\).

Step4: Factor out x

Factor out \(x\) from the left - hand side: \(x(5\ln(2)-\ln(3))=-3\ln(3)-4\ln(2)\).

Step5: Solve for x

Divide both sides by \((5\ln(2)-\ln(3))\) to solve for \(x\): \(x=\frac{-3\ln(3)-4\ln(2)}{5\ln(2)-\ln(3)}\) or we can factor out the negative sign from the numerator: \(x=\frac{-(3\ln(3)+4\ln(2))}{5\ln(2)-\ln(3)}=\frac{3\ln(3)+4\ln(2)}{\ln(3)-5\ln(2)}\)

Answer:

The solution in terms of natural logarithms is \(x = \frac{-3\ln(3)-4\ln(2)}{5\ln(2)-\ln(3)}\) (or equivalent form \(x=\frac{3\ln(3) + 4\ln(2)}{\ln(3)-5\ln(2)}\))