6.4 Power Law of Logarithms
Chapter
Chapter 6
Section
6.4
Lecture on Power Law and Change of Base Laws 5 Videos
Solutions 38 Videos

Evaluate.

 \displaystyle \log_216^3 

Q1a

Evaluate.

 \displaystyle \log_48^2 

Q1b

Evaluate.

 \displaystyle \log 100^{-4} 

Q1c

Evaluate.

 \displaystyle \log_2\sqrt{8} 

Q2a

Evaluate.

 \displaystyle \log_3\sqrt{243} 

Q2b

Evaluate.

 \displaystyle \log_3{(\sqrt{81})^6} 

Q2c

Evaluate.

 \displaystyle \log_4(\sqrt{15})^{15} 

Q2d

Solve for t to two decimal places.

 \displaystyle 10 = 4^t 

Q3a

Solve for t to two decimal places.

 \displaystyle 5^t = 250 

Q3b

Solve for t to two decimal places.

 \displaystyle 2 = 1.08^t 

Q3c

Solve for t to two decimal places.

 \displaystyle 500 =100(1.06)^t 

Q3d

Evaluate, correct to three decimal places.

 \displaystyle \log_223 

Q5a

Evaluate, correct to three decimal places.

 \displaystyle \log_620 

Q5b

Evaluate, correct to three decimal places.

 \displaystyle \log_72 

Q5c

Evaluate, correct to three decimal places.

 \displaystyle -\log_{12}4 

Q5d

Evaluate, correct to three decimal places.

 \displaystyle -\log_{\frac{1}{2}}30 

Q5e

Evaluate, correct to three decimal places.

 \displaystyle \log_{\frac{3}{4}}8 

Q5f

Write as a single logarithm.

 \displaystyle \frac{\log 8}{\log 5} 

Q6a

Write as a single logarithm.

 \displaystyle \frac{\log 17}{\log 9} 

Q6b

Write as a single logarithm.

 \displaystyle \frac{\log \frac{1}{2}}{\log \frac{2}{3}} 

Q6c

Write as a single logarithm.

 \displaystyle \frac{\log(x+1)}{\log(x-1)} 

Q6d

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.

Q7a

Solve for x, correct to three decimal places.

 \displaystyle 2= \log 3^x 

Q9a

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.
Q10a

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?
Q10b

Does \log(mx) = m\log x for some constant m? Explain why or why not, suing algebraic or graphical reasoning and supporting examples.

Q11

Does \log(x^n) = (\log x)^n for some constant n?

Q12

Evaluate \log_28^5 without using the power law of logarithms.

Q13a

Evaluate \log_28^5 the same expression by applying the power law of logarithms.

Q13b

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?

Q15

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
Q16

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 

Q17

Express the following in terms of base-2 logs.

 \displaystyle \log_39 

Q18a

Express the following in terms of base-2 logs.

 \displaystyle \log_325 

Q18b

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.

Q19

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.