Chapter 3, Problem 41CLT

To determine

To solve the exponential equation:

  $4^{x-5}+21=30$

Round the result to three decimal places and to verify the result graphically.

Answer to Problem 41CLT

6.585

Explanation of Solution

Given:

  $4^{x-5}+21=30$

Concept Used:

  ${\log }_{a}x=\frac{\ln x}{\ln a}$

  $u=a^x\iff {\log }_a\left(u\right)=x$

Calculation:

To solve the exponential equation:

  $4^{x-5}+21=30$

First, simplify the equation as,

  $\begin{array}{c}{4}^{x-5}+21=30\\ \Rightarrow 4^{x-5}=30-21\\ \Rightarrow 4^{x-5}=9\end{array}$

Now rewrite it in logarithmic equation using the property $u=e^x\iff \ln \left(u\right)=x$

As,

  $\begin{array}{c}{4}^{x-5}=9\\ \Rightarrow x-5={\log }_{4}9\\ \Rightarrow x-5=\frac{\ln 9}{\ln 4};\text{ Since }{\log }_{a}x=\frac{\ln x}{\ln a}\\ \Rightarrow x\simeq \frac{2.197}{1.386}+5\\ \Rightarrow x\simeq 1.585+5\\ \Rightarrow x\simeq 6.585\end{array}$

Thus, solution of given equation is approximately 6.585.

Now, to verify this result graphically plot the graph of function $4^{x-5}+21$ and then check whether at x approximately equals to 6.585 value of the function is 30, that is, check whether the point $\left(6.585,30\right)$ lies on the curve of this graph.

  

Since, the point $\left(6.585,30\right)$ lies on the graph of the function $4^{x-5}+21$ , thus the result is verified.