Chapter 11.2, Problem 100S

To determine

To solve the equation, ${16}^{d-4}=3^{3-d}$

Answer to Problem 100S

  $d\approx 3.7162$

Explanation of Solution

Given:

The equation: ${16}^{d-4}=3^{3-d}$

Concept Used:

Take log both sides and use the rule of logarithm: $\log a^x=x\log a$

Calculation:

In order to solve the given equation, take natural logarithm both sides and simplify further as shown below

  $\begin{array}{c}{16}^{d-4}=3^{3-d}\\ \ln {16}^{d-4}=\ln 3^{3-d}\\ \left(d-4\right)\ln 16=\left(3-d\right)\ln 3\\ d\ln 16-4\ln 16=3\ln 3-d\ln 3\\ d\ln 16+d\ln 3=3\ln 3+4\ln 16\\ d\left(\ln 16+\ln 3\right)=3\ln 3+4\ln 16\\ d=\frac{3\ln 3+4\ln 16}{\ln 16+\ln 3}\\ d\approx 3.7162\end{array}$

Thus, the solution of the equation to the nearest ten-thousand is $d\approx 3.7162$