2.1 Change in Arithmetic and Geometric Sequences: แบบฝึกหัด

Which of the following is the exponential form of the logarithmic equation \( \log_{b}(y) = x \)?

Consider the function \( f(x) = a \cdot b^x + c \). If the graph of \( f \) has a horizontal asymptote at \( y = -4 \) and passes through the points \( (0, 2) \) and \( (1, 8) \), what is the value of \( f(2) \)?

Find the value of \( x \) that satisfies the equation \( 4^{x-1} = 8^{2x+3} \).

Evaluate the logarithmic expression \( \log_2(32) \).

The intensity of an earthquake is measured on the Richter scale using the formula \( R = \log\left(\frac{I}{I_0}\right) \), where \( I \) is the intensity and \( I_0 \) is a constant. If Earthquake A has a Richter scale magnitude of 6.2 and Earthquake B is 40 times as intense as Earthquake A, what is the magnitude of Earthquake B? (Round to the nearest tenth)

A population of a certain bacteria culture follows the growth model \( N(t) = N_0 e^{kt} \), where \( t \) is the time in hours.
(a) If the population triples every 4 hours, find the exact value of the growth constant \( k \).
(b) Determine the time \( t \) required for the population to reach 5 times its initial amount. Round your answer to the nearest hundredth of an hour.