Question

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.

$4^x-10·4^{-x}=3$

Short answer

The exact solution is ${\log }_{4}5$and the rounded solution is $1.161$.

Step-by-step solution

Step 1. Rewrite the given equation.

The given equation is:

$4^x-10·4^{-x}=3$

It can be rewritten as:

$4^x-\frac{10}{4^x}=3$

Multiply both sides by $4^x$.

$\begin{array}{rcl}(4^x{)}^2-10& =& 3\left(4^x\right)\\ (4^x{)}^2-3(4^x)-10& =& 0\end{array}$

Step 2. Substitute another variable and solve for it.

Let $4^x=u$.

$\begin{array}{rcl}{u}^2-3u-10& =& 0\\ u^2-5u+2u-10& =& 0\\ u(u-5)+2(u-5)& =& 0\\ (u+2)(u-5)& =& 0\end{array}$

Use the zero product property.

$\begin{array}{rcl}u+2& =& 0\text{or}u-5=0\\ u& =& -2\text{or}u=5\end{array}$

Step 3. Determine the solution.

Substitute $u=4^x$in the above solutions.

$4^x=-2\text{or}{4}^x=5$

The equation $4^x=-2$has no solution because $4^x>0$. Now, solve localid="1648448652103" $4^x=5$.

localid="1648449962198" $\begin{array}{rcl}\ln 4^x& =& \ln 5\\ x\ln 4& =& \ln 5\\ x& =& \frac{\ln 5}{\ln 4}\\ x& =& {\log }_{4}5\\ x& \approx & 1.161\end{array}$

The exact solution is ${\log }_{4}5$and the rounded solution is $1.161$.

Step 4. Verify the result by the graphing utility.

Plot the graphs of $y_1=4^x-10·4^{-x}$ and$y_2=3$ on a coordinate plane. The x-coordinate of the intersection point is the solution of the given equation.

The graphs intersect each other at $x\approx 1.161$. Hence, the solution is verified.