Exponential and Logarithmic Functions

Simplify the logarithmic expression \( log_{3}(9) - log_{3}(\sqrt{3}) \)

Expand the logarithm expression \(\log_{2}(16x^3)\)

Consider the function $f(x)=e^{x}$. a. Differentiate the Taylor series about 0 of $f(x)$. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative.

In 2004 , an art collector paid $\$ 84,053,000$ for a particular painting. The same painting sold for $\$ 30,000$ in 1950. Complete parts (a) through (d). $V(t)=30000 \times e^{0.147 t}$ (Type an expression. Type integers or decimals for any numbers in the expression. Round to three decimal places as needed.) b) Predict the value of the painting in 2024 . $\$ 1,590,000,000$ (Round to the nearest million as needed.) c) Estimate the rate of change of the painting's value in 2024 . dollar(s) per year. (Round to the nearest million as needed.)