Exponential and Logarithmic Functions

The general form of an exponential function can be expressed as y=a*b^x, in which 'a' represents a constant, 'b' identifies the base, and 'x' assumes the role of the exponent. Conversely, logarithmic functions, serving as the inverse of exponential functions, are represented by the equation y=log_b(x), where 'b' denotes the base. These particular functions play a pivotal role in various fields of mathematics and applied sciences.

Simplifying Logarithmic Expressions

Simplify the logarithmic expression log3(9)log3(3)

Expanding Logarithmic Expressions

Expand the logarithm expression log2(16x3)

Exponential Expressions

Consider the function f(x)=ex. 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.

Exponential Equations

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×e0.147t (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.)