Chapter 3, Problem 42CLT

To determine

To solve the exponential equation:

  ${\log }_{2}x+{\log }_{2}5=6$

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

Answer to Problem 42CLT

6.585

Explanation of Solution

Given:

  ${\log }_{2}x+{\log }_{2}5=6$

Concept Used:

  ${\log }_{a}x+{\log }_{a}y={\log }_a\left(xy\right)$

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

Calculation:

To solve the exponential equation:

  ${\log }_{2}x+{\log }_{2}5=6$

First, simplify the equation using property of logarithmic function, as

  $\begin{array}{c}{\log }_{2}x+{\log }_{2}5=6\\ \Rightarrow {\log }_2\left(5x\right)=6\end{array}$

Now rewrite it in exponential equation using the property $u=a^x\iff {\log }_a\left(u\right)=x$

As,

  $\begin{array}{c}{\log }_2\left(5x\right)=6\\ \Rightarrow 5x=2^6\\ \Rightarrow x=\frac{64}{5}\\ \Rightarrow x=12.8\end{array}$

Thus, solution of given equation is approximately 12.8.

Now, to verify this result graphically plot the graph of function ${\log }_{2}x+{\log }_{2}5$ and then check whether at x equals to 12.8 value of the function is 6, that is, check whether the point $\left(12.8,6\right)$ lies on the curve of this graph.

  

Since, the point $\left(12.8,6\right)$ lies on the graph of the function ${\log }_{2}x+{\log }_{2}5$ , thus the result is verified.