Question

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places

${\left(\frac{1}{3}\right)}^x=7$

Short answer

The approximate value of x for the equation ${\left(\frac{1}{3}\right)}^x=7$is $-1.771$.

Step-by-step solution

Step 1. Given data 

The given Logarithmic Equation is${\left(\frac{1}{3}\right)}^x=7$.

Step 2. Use logarithms  

Take natural logarithms to both sides of the equation

$$\begin{gathered} {\left(\frac{1}{3}\right)}^x=7 \\ \ln {\left(\frac{1}{3}\right)}^x=\ln 7 \end{gathered}$$

Step 3. Apply power property of logarithm

Use power property of logarithm

$$\begin{gathered} \ln {\left(\frac{1}{3}\right)}^x=\ln 7 \\ x\ln \left(\frac{1}{3}\right)=\ln 7 \\ x(\ln 1-\ln 3)=\ln 7 \\ x(0-1.09)=1.94 \\ x=-\frac{1.94}{1.09} \\ x=-1.77 \end{gathered}$$