Ch. 8 Logarithmic Functions
Lesson 1 Understanding Logarithms
Logarithm (log) is just another representation for the value of an exponent. It represents the exponent, the power to which the base must be raised to produce a given number.
For example, log10 100 means: à 10 (base) to the power of what value gives 100 ?
**As base 10 is the most common logarithm in science, it is usually written as log without the 10.
Ex. Evaluate the following:
a) b) c)
d) e) f)
Definition of Logarithm:çè
**NOTE: Your calculator reads “log” as “”.
Ex. Rewrite each of the following from exponential form to logarithmic form.
a) b)
Ex. Rewrite each of the following from logarithmic form to exponential form.
a) b)
Ex. Evaluate the following.
a) b) c)
d) e) f)
g) h) i)
General idea on domain & range:
Classwork (Optional) Definition of Logarithms
1. Evaluate.
a. b. c.
d. e. f.
2. Solve each equation for x.
a. b.
c. d.
Answers:
1. a) 8 b) 8 c) d) e) f) 1 2. a) 100 b) 64 c) 9 d)
Ex. Using table of values, graph & on the same grid.
: / Domain / : / Domain
Range / Range
Asymptote / Asymptote
Therefore, what is the relationship between the exponential function logarithmic function (with the same base)?
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Ch. 8 Logarithmic Functions
Lesson 2 Transformations of Logarithmic Functions
Just as before, we can transform any logarithmic function as follows:
Ex. a) Use transformations to sketch the graph of the function .
b) Identify the following characteristics of the graph of the function:
i) the equation of the asymptote ii) the domain and range
iii) the y-intercept iv) the x-intercept
Ex. a) Use transformations to sketch the graph of the function.
b) Identify the following characteristics of the graph of the function:
i) the equation of the asymptote ii) the domain and range
iii) the y-intercept iv) the x-intercept
Ex. The graph shown can be generated by stretching the graph of . Write the equation that describes the graph.
Ex. Welders wear helmets fitted with a filter shade to protect their eyes from intense light. The filter shade number, N, is defined by the function , where T is the fraction of visible light that passes through the filter. Shade numbers range from 2 to 14, with a lens shade number of 14 allowing the least amount of light to pass through. What fraction of light is passed through a filter of shade numbered 12?
Ch. 8 Logarithmic Functions
Lesson 3 Laws of Logarithms
Warm-up: Evaluate or solve the following:
a) b) c)
Laws of Logarithms
Multiplication Law: = or =
Division Law: = or =
Power Law: = or =
Ex. Simplify the following into a single logarithmic form.
a) log 5 + log 12 b) log6 10 + log6 18 – log6 5
c) log a + log b – log c + log d d) 2 log4 x – log4 (x + 3)
.
Ex. Write as a single logarithm.
a) b)
Ex. Evaluate.
a) b)
Ex. If and , evaluate .
Ex. If log 2 = & log 3 = , express the following in terms of & .
a) log 72 b) log 600
Change of Base:
where n is any base ( & )
Ex. Evaluate each logarithm to 3 decimal places using technology.
a) b)
Ex. The pH scale is used to measure the acidity or alkalinity of a solution. The pH of a solution is defined as pH = -log[H+], where [H+] is the hydrogen ion concentration in mol/L. A neutral solution, such as water, has a pH of 7, acids have pH < 7, and bases have pH > 7.
a) If cola has a pH of 2.5, how much more acidic is it than water?
b) An apple is 5 times as acidic as a pear. If a pear has a pH of 3.8, then what is the pH of an apple?
Ch. 8 Logarithmic Functions
Lesson 4 (Part I) Solving Logarithmic Equations
Ex. State the restrictions on x for each of the following equations:
a) b)
When solving equations involving logarithms, there are two general forms:
1) If you can simplify both sides to a single logarithm with the same base, then you can equate the contents of the logarithms!
2) If you have a logarithm equal to a numerical value, re-write in exponential form!
Note: When giving final solutions, we must always be aware of restrictions!
Ex. Solve each of the following and check your solution(s).
a) b)
c) d)
.
Ex. Solve.
a) b)
Lesson 4 (Part II) Solving Logarithmic Equations & Applications
If we are solving an exponential equation that cannot produce matching bases, then we can use logarithms: Power Rule allows us to take the exponent out in front and solve!
Ex. Solve for x:
a) b)
c)
Ex. Strontium-90 has a half-life of 28 years. How long will it take an 85-gram sample to decay to 15 g?
Ex. How long would it take a population of bees to triple if they multiply 8 fold every 5 weeks?
Ex. If an earthquake registers 2.7, how high on the Richter Scale does an earthquake 52 000 times stronger register at?
Ex. Calculate the number of years for an investment of $1000 to triple at an interest rate of 6.5% compounded semi-annually. (Use “compound interest” formula).
Ex. 20% of a certain radioactive isotope decays in 7 days. What is the half-life of the isotope?