91Ó°ÊÓ

Can the natural base \(e\) be written as a ratio of two integers? Explain.

Solve the equation. Check for extraneous solutions. \(\log _3(x-9)+\log _3(x-3)=2\)

Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(f(x)=\log _6 x\); vertical stretch by a factor of 6 , followed by a translation 5 units down

Use the change-of-base formula to evaluate the logarithm. $$\log _7 \frac{3}{16}$$

You invest \(\$ 500\) in the stock of a company. The value of the stock decreases \(2 \%\) each year. Describe and correct the error in writing a model for the value of the stock after \(t\) years.

Simplify the expression. \(\log _3 3^{2 x}\)

You plant a sunflower seedling in your garden. The height \(h\) (in centimeters) of the seedling after \(t\) weeks can be modeled by the logistic function $$ h(t)=\frac{256}{1+13 e^{-0.65 t}} . $$ a. Find the time it takes the sunflower seedling to reach a height of 200 centimeters. b. Use a graphing calculator to graph the function. Interpret the meaning of the asymptote in the context of this situation.

Your friend evaluates \(f(x)=e^{-x}\) when \(x=1000\) and concludes that the graph of \(y=f(x)\) has an \(x\)-intercept at \((1000,0)\). Is your friend correct? Explain your reasoning.

Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(f(x)=\log _5 x\); reflection in the \(x\)-axis, followed by a translation 9 units left

Solve the equation. Check for extraneous solutions. \(\log _5(x+4)+\log _5(x+1)=2\)