Question

Consider the following.10^(1 − x )= 6^x(a) Find the exact solution of the exponential equation in terms of logarithms.x = (b) Use a calculator to find an approximation to the solution rounded to six decimal places.x =

Answer 1

We have to find the exact solution to the equation.

We can solve it as:

`\begin{gathered} 10^(1-x)=6^x \\ (1-x)\cdot\ln10=x\cdot\ln(6) \\ \ln10-\ln10\cdot x=\ln6\cdot x \\ \ln10=(\ln6-\ln10)x \\ x=(\ln10)/(\ln6-\ln10) \end{gathered}`

b) If we use a calculator to find the logarithms we obtain the approximate value of x as:

`\begin{gathered} \ln10\approx2.3025851 \\ \ln6\approx1.7917595 \\ \Rightarrow x\approx(2.3025851)/(1.7917595-2.3025851)=(2.3025851)/(-0.5108256)\approx-4.507576 \end{gathered}`

Answer:

a) x = ln(10)/(ln(6)-ln(10))

b) x = -4.507576