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
\frac{-3\ln(3) - 4\ln(2)}{5\ln(2) - \ln(3)}
(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.)
now use a calculator to obtain a decimal approximation for the solution.
the solution set is {}.
(use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Recall logarithm properties

We have the equation \(2^{5x + 4}=3^{x - 3}\). Take the natural logarithm of both sides: \(\ln(2^{5x + 4})=\ln(3^{x - 3})\).
Using the property \(\ln(a^b)=b\ln(a)\), we get \((5x + 4)\ln(2)=(x - 3)\ln(3)\).

Step2: Expand both sides

Expand the equation: \(5x\ln(2)+4\ln(2)=x\ln(3)-3\ln(3)\).

Step3: Collect like terms

Move terms with \(x\) to one side and constants to the other: \(5x\ln(2)-x\ln(3)=- 3\ln(3)-4\ln(2)\).
Factor out \(x\): \(x(5\ln(2)-\ln(3))=-3\ln(3)-4\ln(2)\).

Step4: Solve for \(x\)

Divide both sides by \(5\ln(2)-\ln(3)\): \(x = \frac{-3\ln(3)-4\ln(2)}{5\ln(2)-\ln(3)}\).

Step5: Calculate the decimal approximation

First, calculate the numerator: \(-3\ln(3)-4\ln(2)\approx-3\times1.0986 - 4\times0.6931=-3.2958 - 2.7724=-6.0682\).
Then, calculate the denominator: \(5\ln(2)-\ln(3)\approx5\times0.6931 - 1.0986 = 3.4655-1.0986 = 2.3669\).
Now, divide numerator by denominator: \(x\approx\frac{-6.0682}{2.3669}\approx - 2.56\).

Answer:

The solution in terms of natural logarithms is \(\frac{-3\ln(3)-4\ln(2)}{5\ln(2)-\ln(3)}\) and the decimal approximation is \(-2.56\) (rounded to two decimal places).