Exponential and logarithmic functions and Graphing techniques: แบบฝึกหัด

Using the laws of logarithms, simplify the expression \(2 \ln a - \ln b\).

The graph of the function \(y = \ln(3 - x) + 2\) has a vertical asymptote. What is the equation of this asymptote?

Using a graphing calculator or software, determine the number of real roots for the equation \( e^x = 4 - x^2 \).

Solve the equation \(\ln x + \ln 2 = \ln 10\) for \(x\).

Consider the graph of the function \( y = \ln(x - 2) \). What is the equation of its vertical asymptote?

Solve the equation \( \ln(2x - 1) = 0 \).

State the equations of the vertical and horizontal asymptotes for the graph of \( y = \ln(x + 3) - 4 \).

Solve the logarithmic equation \(2\log_{10}(x) - \log_{10}(x + 4) = \log_{10}(2)\), giving your answer(s) to 3 decimal places where necessary.

Consider the function \(y = 3 - e^{-2x}\).
(a) State the equation of the horizontal asymptote of the graph of the function.
(b) Find the coordinates of the point where the graph crosses the \(x\)-axis.
(c) Sketch the graph of the function, labeling the intercept found in part (b) and the horizontal asymptote.