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 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 \(2^{5x + 4}=3^{x - 3}\). We get \(\ln(2^{5x + 4})=\ln(3^{x - 3})\).

Step2: Apply logarithm power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), we can rewrite the equation as \((5x + 4)\ln(2)=(x - 3)\ln(3)\).

Step3: Expand both sides

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

Step4: Collect like terms

Move all terms with \(x\) to one side and the constant terms to the other side. \(5x\ln(2)-x\ln(3)=- 3\ln(3)-4\ln(2)\).

Step5: Factor out x

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

Step6: Solve for x

Solve for \(x\) by dividing both sides by \((5\ln(2)-\ln(3))\): \(x=\frac{-3\ln(3)-4\ln(2)}{5\ln(2)-\ln(3)}=\frac{3\ln(3)+4\ln(2)}{\ln(3)-5\ln(2)}\) (we can multiply the numerator and denominator by - 1).

To find the decimal approximation, we know that \(\ln(2)\approx0.6931\), \(\ln(3)\approx1.0986\).

First, calculate the numerator: \(3\ln(3)+4\ln(2)=3\times1.0986 + 4\times0.6931=3.2958+2.7724 = 6.0682\)

Then, calculate the denominator: \(\ln(3)-5\ln(2)=1.0986-5\times0.6931=1.0986 - 3.4655=-2.3669\)

So, \(x=\frac{6.0682}{-2.3669}\approx - 2.564\)

Answer:

In terms of logarithms: \(x = \frac{\ln(3^{3}\times2^{4})}{\ln(3)-\ln(2^{5})}=\frac{\ln(27\times16)}{\ln(3)-\ln(32)}=\frac{\ln(432)}{\ln(\frac{3}{32})}\) (or \(x=\frac{3\ln(3)+4\ln(2)}{\ln(3)-5\ln(2)}\))

Decimal approximation: \(x\approx - 2.56\) (rounded to two decimal places)