Question

solve the equation. give an exact solution, and also an approximate solution to four decimal places. 2^{4x - 9}=11 a. the exact solution is x = . (use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Take the logarithm of both sides

Take the natural - logarithm of both sides of the equation $2^{4x - 9}=11$. We get $\ln(2^{4x - 9})=\ln(11)$.

Step2: Use the power - rule of logarithms

By the power - rule of logarithms $\ln(a^b)=b\ln(a)$, the left - hand side becomes $(4x - 9)\ln(2)=\ln(11)$.

Step3: Solve for x

First, divide both sides by $\ln(2)$: $4x-9=\frac{\ln(11)}{\ln(2)}$. Then, add 9 to both sides: $4x=\frac{\ln(11)}{\ln(2)} + 9$. Finally, divide by 4: $x=\frac{\frac{\ln(11)}{\ln(2)}+9}{4}=\frac{\ln(11)+9\ln(2)}{4\ln(2)}$.

Step4: Calculate the approximate value

We know that $\ln(11)\approx2.3979$ and $\ln(2)\approx0.6931$.
\[

$$\begin{align*} x&=\frac{2.3979 + 9\times0.6931}{4\times0.6931}\\ &=\frac{2.3979+6.2379}{2.7724}\\ &=\frac{8.6358}{2.7724}\\ &\approx3.1149 \end{align*}$$

\]

Answer:

The exact solution is $x = \frac{\ln(11)+9\ln(2)}{4\ln(2)}$, and the approximate solution to four decimal places is $x\approx3.1149$.