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.
7^{2x + 3} = 3^{x - 4}
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 log on both sides

Take the natural logarithm (ln) of both sides of the equation \(7^{2x + 3}=3^{x - 4}\). We get \(\ln(7^{2x + 3})=\ln(3^{x - 4})\).

Step2: Apply log power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), we can rewrite the equation as \((2x + 3)\ln(7)=(x - 4)\ln(3)\).

Step3: Expand both sides

Expand the left - hand side and the right - hand side: \(2x\ln(7)+3\ln(7)=x\ln(3)-4\ln(3)\).

Step4: Collect x 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 \(3\ln(7)\) from both sides: \(2x\ln(7)-x\ln(3)=- 4\ln(3)-3\ln(7)\).

Step5: Factor out x

Factor out \(x\) from the left - hand side: \(x(2\ln(7)-\ln(3))=-4\ln(3)-3\ln(7)\).

Step6: Solve for x

Divide both sides by \((2\ln(7)-\ln(3))\) to solve for \(x\): \(x=\frac{-4\ln(3)-3\ln(7)}{2\ln(7)-\ln(3)}\) or we can factor out the negative sign from the numerator: \(x=\frac{-(4\ln(3)+3\ln(7))}{2\ln(7)-\ln(3)}=\frac{4\ln(3)+3\ln(7)}{\ln(3)-2\ln(7)}\) (by multiplying numerator and denominator by - 1)

Answer:

\(x = \frac{-4\ln(3)-3\ln(7)}{2\ln(7)-\ln(3)}\) (or equivalent form \(\frac{4\ln(3)+3\ln(7)}{\ln(3)-2\ln(7)}\))