11.6 Exponential and Logarithmic Functions

Simplify $x^{ 5} \cdot x^{ 3}$ as a single power of $x$.

Simplify $\dfrac{x^{ 8}}{x^{ 3}}$ as a single power of $x$.

Simplify $\left(z^{ 4}\right)^{ 3}$ as a single power.

Write $\left(\dfrac{1}{2}\right)^{-3}$ as a whole number.

Evaluate $(17)^0$.

State the $y$-intercept and the equation of the horizontal asymptote of $y = 2^x$.

Evaluate $ 2^{ 6}$.

Evaluate $ 49^{1/2}$ exactly.

Solve $ 2^x = 64$, giving an exact answer.

CHF\,1000 is invested at 5% per year, compounded annually. Find the value after 1 year.

Write $ 2^{ 5} = 32$ in logarithmic form.

Evaluate $\log_{ 5} 125$.

A radioactive substance decays at 12% per year. Find the multiplier per year, and the amount after 5 years if initial mass is 100 g.

Evaluate $ 8^{2/3}$ exactly.

A colony of 200 bacteria doubles every hour. Find the population after 4 hours.

Simplify $\log 8 + \log 5 - \log 4$.

A function $y = a \cdot b^x$ has $y$-intercept 5 and passes through $(1, 15)$. Find $a$ and $b$.

A car depreciates at 18% per year. If the new price is CHF 30 000, find the value after 3 years.

Express $\log_a (p^3 q^2) - \log_a(p q^3)$ in terms of $\log_a p$ and $\log_a q$.

A population is modelled by $P(t) = 500 \cdot (1.04)^t$ where $t$ is years. State the meaning of the values 500 and 1.04.

CHF 5000 is invested at 4.5% per year, compounded annually. Find the value after 8 years, to the nearest franc.

Solve $ 5^x = 17$, giving the answer correct to 3 s.f.

Describe the transformation from $y = 2^x$ to $y = 2^{x - 3} + 1$.

Solve $\log_3 x + \log_3(x - 2) = 1$.

A model $C(t) = a \cdot b^t$ satisfies $C(1) = 6$ and $C(4) = 162$. Find $a$ and $b$.

How many full years does it take for CHF 1000 to grow to CHF 2000 at 5% per year compounded annually?

A population of bacteria triples every 2 hours. If $N(0) = 200$, write a model $N(t) = a \cdot b^t$.

A model $M(t) = 80 \cdot 0.85^t$ describes the temperature drop of a coffee. Find (a) $M(0)$, (b) $M(10)$ to 1 d.p., (c) the long-run behaviour.

A function is $y = 3 \cdot 2^{x - 1} + 2$. State (a) the $y$-intercept, (b) the asymptote, (c) whether it grows or decays.

Compare: (a) CHF 1000 at 5% compounded annually for 10 years; (b) CHF 1000 at 4.9% compounded monthly for 10 years. Which gives more, and by how much (to nearest franc)?

Solve $4 \cdot 2^{2x} - 9 \cdot 2^x + 2 = 0$ for real $x$.

Solve $\log_2 x - \log_2(x - 3) = 2$, giving an exact answer.

A population follows $P(t) = 200 \cdot 3^t$ weeks. Find the smallest $t$ (to 3 s.f.) for which $P > 100\,000$.

An investment doubles in 8 years with annual compounding. Find the annual interest rate $r$ (to 3 s.f.).

A bacteria colony is modelled by $N(t) = a \cdot e^{kt}$. At $t = 0$, $N = 200$; at $t = 5$, $N = 800$. Find $a$ and $k$ (to 3 s.f.).

A radioactive isotope has a half-life of 12 years. Find the proportion remaining after 30 years (to 3 d.p.).

Evaluate $\log_4 32$ exactly using the change-of-base law.

The graph of $y = a \cdot 2^x + c$ has $y$-intercept $(0, 4)$ and horizontal asymptote $y = -3$. Find $a$ and $c$.

A model $h(t) = h_0 \cdot r^t$ describes the bounce height of a ball. After bounce 1, height is 80 cm; after bounce 4, height is 32.768 cm. Find $h_0$ and $r$.

Solve $2^{x + 1} = 5 \cdot 3^x$ to 3 s.f.