Question

1. which of the following is the graph of the logarithmic function $f(x) = \log_{2}(\frac{1}{2}x + 2) + 4$?

a) graph
b) graph
c) graph
d) graph

1. on a remote island, the fish population has been declining by 8% per year. the current population is 2,000 fish. assume that the fish population continues to decrease by 8% per year. which of the following logarithmic equations best models the number of years it will take for the fish population to decrease from 2,000 fish to 1,500 fish?

a) $t = \log_{0.75} 1.08$
b) $t = \log_{0.75} 0.92$
c) $t = \log_{1.08} 0.75$
d) $t = \log_{0.92} 0.75$

1. on a secluded island, the turtle population has been decreasing by 4% annually. the current population is 5,000 turtles. assuming the turtle population continues to decline by 4% each year, which of the following logarithmic equations best represents the number of years it will take for the turtle population to reduce from 5,000 turtles to 3,000 turtles?

a) $t = \log_{0.6} 0.96$
b) $t = \log_{1.04} 0.6$
c) $t = \log_{0.6} 1.04$
d) $t = \log_{0.96} 0.6$

Explanation:

Step1: Find vertical asymptote

For $f(x)=\log_2(x+2)+4$, set $x+2=0$ → $x=-2$.

Step2: Find key point

Set $x+2=1$ → $x=-1$, then $f(-1)=0+4=4$. So point $(-1,4)$.

Step3: Match graph

Check graphs: only option b has asymptote $x=-2$ and passes through $(-1,4)$.

Step1: Decay formula

$P(t)=P_0(1-r)^t$, where $P_0=2000$, $r=0.08$, $P(t)=1500$.

Step2: Substitute values

$1500=2000(0.92)^t$ → $\frac{1500}{2000}=0.92^t$ → $0.75=0.92^t$.

Step3: Convert to log

$t=\log_{0.92}0.75$.

Step1: Decay formula

$P(t)=P_0(1-r)^t$, $P_0=5000$, $r=0.04$, $P(t)=3000$.

Step2: Substitute values

$3000=5000(0.96)^t$ → $\frac{3000}{5000}=0.96^t$ → $0.6=0.96^t$.

Step3: Convert to log

$t=\log_{0.96}0.6$.

Answer:

b. The graph with asymptote x=-2 and point (-1,4)