91Ó°ÊÓ

Solve the equation.\(5^{2 x}+20 \cdot 5^x-125=0\)

In Exercises 67-70, you solved exponential and logarithmic equations with different bases. Describe general methods for solving such equations.

Let \(f(x)=\sqrt[3]{x}\). Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(g(x)=f\left(\frac{1}{2} x\right)\)

When X-rays of a fixed wavelength strike a material \(x\) centimeters thick, the intensity \(I(x)\) of the X-rays transmitted through the material is given by \(I(x)=I_0 e^{-\mu x}\), where \(I_0\) is the initial intensity and \(\mu\) is a value that depends on the type of material and the wavelength of the X-rays. The table shows the values of \(\mu\) for various materials and X-rays of medium wavelength. $$ \begin{array}{|l|c|c|c|} \hline \text { Material } & \text { Aluminum } & \text { Copper } & \text { Lead } \\ \hline \text { Value of } \mu & 0.43 & 3.2 & 43 \\ \hline \end{array} $$a. Find the thickness of aluminum shielding that reduces the intensity of \(\mathrm{X}\)-rays to \(30 \%\) of their initial intensity. (Hint: Find the value of \(x\) for which \(I(x)=0.3 I_0\). b. Repeat part (a) for the copper shielding. c. Repeat part (a) for the lead shielding. d. Your dentist puts a lead apron on you before taking X-rays of your teeth to protect you from harmful radiation. Based on your results from parts (a)-(c), explain why lead is a better material to use than aluminum or copper.

Let \(f(x)=\sqrt[3]{x}\). Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(g(x)=f(-x)+3\)

Write an equation in point-slope form of the line that passes through the given point and has the given slope.\((1,-2) ; m=4\)

Let \(f(x)=\sqrt[3]{x}\). Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(g(x)=f(x+2)\)

Write an equation in point-slope form of the line that passes through the given point and has the given slope. \((3,2) ; m=-2\)

Write an equation in point-slope form of the line that passes through the given point and has the given slope.\((3,-8) ; m=-\frac{1}{3}\)

Write an equation in point-slope form of the line that passes through the given point and has the given slope.\((2,5) ; m=2\)