1 Log Law 3 Power Law
2 Change of Bases Law and Proof
3 Change of Base Ex1
4 Change of Base Ex2
Evaluate.
\displaystyle
\log_216^3
Evaluate.
\displaystyle
\log_48^2
Evaluate.
\displaystyle
\log 100^{-4}
Evaluate.
\displaystyle
\log_2\sqrt{8}
Evaluate.
\displaystyle
\log_3\sqrt{243}
Evaluate.
\displaystyle
\log_3{(\sqrt[3]{81})^6}
Evaluate.
\displaystyle
\log_4(\sqrt[5]{15})^{15}
Solve for t to two decimal places.
\displaystyle
10 = 4^t
Solve for t to two decimal places.
\displaystyle
5^t = 250
Solve for t to two decimal places.
\displaystyle
2 = 1.08^t
Solve for t to two decimal places.
\displaystyle
500 =100(1.06)^t
Evaluate, correct to three decimal places.
\displaystyle
\log_223
Evaluate, correct to three decimal places.
\displaystyle
\log_620
Evaluate, correct to three decimal places.
\displaystyle
\log_72
Evaluate, correct to three decimal places.
\displaystyle
-\log_{12}4
Evaluate, correct to three decimal places.
\displaystyle
-\log_{\frac{1}{2}}30
Evaluate, correct to three decimal places.
\displaystyle
\log_{\frac{3}{4}}8
Write as a single logarithm.
\displaystyle
\frac{\log 8}{\log 5}
Write as a single logarithm.
\displaystyle
\frac{\log 17}{\log 9}
Write as a single logarithm.
\displaystyle
\frac{\log \frac{1}{2}}{\log \frac{2}{3}}
Write as a single logarithm.
\displaystyle
\frac{\log(x+1)}{\log(x-1)}
Sketch a graph of the function f(x) = \log_2x by first graphing the inverse function f^{-1}(x) = 2^x and then reflecting the graph in the line y =x.
Solve for x, correct to three decimal places.
\displaystyle
2= \log 3^x
An investment earns 9% interest, compounded annually, The amount, A , that the investment is worth as a function of time, t, in years, is given by `A(t)= 400(1.09)^t
- What was the initial value of the investment? Explain how you know.
An investment earns 9% interest, compounded annually, The amount, A , that the investment is worth as a function of time, t, in years, is given by `A(t)= 400(1.09)^t
- How long will it take for the investment to double in value?
Does \log(mx) = m\log x for some constant m? Explain why or why not, suing algebraic or graphical reasoning and supporting examples.
Does \log(x^n) = (\log x)^n for some constant n?
Evaluate \log_28^5 without using the power law of logarithms.
Evaluate \log_28^5 the same expression by applying the power law of logarithms.
A spacecraft is approaching a space station that is orbiting Earth. When the craft is 1000 km from the space station, reverse thrusters must be applied to begin braking. The time, t, in hours, required to reach a distance, d, in kilometres, from the space station while the thrusters are being
fired can be modelled by t = \log_{0.5}(\frac{d}{1000}).
The docking sequence can be initialized once the craft is within 10 km of the station’s docking bay.
a) How long after the reverse thrusters are first fired should docking procedures begin?
b) What are the domain and range of this function? What do these features represent?
An investment pays 8% interest, compounded annually.
a) Write an equation that expresses the amount, A, of the investment as a function of time, t, in years.
b) Determine how long it will take for this investment to
- i) double in value
- ii) triple in value
c) Determine the percent increase in value of the account after
- i) 5 years
- ii) 10 years
Use algebraic reasoning to show that any logarithm can be written in terms of a logarithm with any base:
\displaystyle
\log_b x = \frac{\log_k x}{\log_k b} \text{ for any } x > 0, k>0, b> 0
Express the following in terms of base-2 logs.
\displaystyle
\log_39
Express the following in terms of base-2 logs.
\displaystyle
\log_325
A computer design contains 10 binary digits in 64 sequences. As a result, the number of codes in (2^{10})^{64}. Express this number in logarithmic form with base 2.
An investment pays 3.5% interest, compounded quarterly.
a) Write an equation to express the amount, A, of the investment as a function of time, t, in years.
b) Determine how long it will take for this investment to
- i) double in value.
- ii) triple in value
Given that f(x) = \frac{1}{\sqrt{1- x }} and f(a) = 2, what is the value of f(1-a)?
Given that e and f are integers, both greater than one, and \sqrt{e\sqrt{e\sqrt{e}}} =f, determine the least possible value of e + f.