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I can express a logarithmic equation as an exponential equation
and vice versa. I can determine the value of a logarithmic
expression, such as, without technology. I can determine the value
of a logarithmic expression, such as, with technology. I can solve
problems that involve logarithmic scales, such as the Richter scale
and the pH scale.
Slide 3
Explore Logarithms are an alternate way of expressing
exponents. The table below shows the relationship between
exponential and logarithmic forms. Study the completed first three
rows. Complete the table by filling in the blanks. Try this on your
own first!
Slide 4
You Should Notice There is a pattern to follow when converting
between exponential and logarithmic form.
Slide 5
Information
Slide 6
Example 1 Convert the exponential equations into logarithmic
equations, and the logarithmic equations into exponential
equations. a) b) c) d) Converting between exponential and
logarithmic forms
Slide 7
Example 2 Evaluate each logarithmic expression without using
technology. a) b) c)d) Evaluating and interpreting a logarithmic
expression without technology = x Since 2 to the power of 4 gives
us 16, x = 4. = x Since the answer is unknown, we can use x as a
placeholder to represent the answer. Then we can change the
expressions to exponential form. Last, we use reasoning to figure
out xs value. Since 4 to the power of 3 gives us 16, x = 3.
Slide 8
Example 2 e) f) = x Since the answer is unknown, we can use x
as a placeholder to represent the answer. Then we can change the
expressions to exponential form. Last, we use reasoning to figure
out xs value. There is no number that can, when placed as an
exponent on 2, give an answer of -4. No Solution.
Slide 9
Example 3 Evaluate each logarithmic equation without using
technology. a) b) c) d) Solving a logarithmic equation without
technology Start by changing the expressions to exponential form.
If it is easy to calculate, do that. x = 7 In cases like these last
two, you can take the root of both sides to isolate x.
Slide 10
Example 4 Describe how to estimate the value following without
using technology. a) b) Estimating the value of a logarithmic
expressions without technology Since the answer is unknown, we can
use x as a placeholder to represent the answer. Then we can change
the expressions to exponential form. Last, we use reasoning to
figure out an approximate value for x. = x A good approximate
answer is x = 3.9.A good approximate answer is x = 2.6.
Slide 11
Example 5 Evaluate each logarithmic expression using
technology. a) b) Evaluating and interpreting a logarithmic
expression with technology To calculate these in your calculator,
use the change of base formula. Keep in mind that on most
calculators, the log button is base 10.
Slide 12
Example 6 Using logarithms to solve a contextual problem
Slide 13
Example 7 The pH, is defined by the equation pH = log C, where
the concentration of hydrogen ions, C, in a solution is measured in
moles per litre (mol/L). Using logarithms to determine and compare
pH levels
Slide 14
Example 7 a)The hydrogen ion concentration, C, of a solution is
0.0001 mol/L. Calculate the pH of the solution. b) Use the pH scale
shown in the diagram above to calculate the hydrogen ion
concentration of lemon juice. Looking at the table, Divide both
sides by -1. Change to exp. form.
Slide 15
Example 7 c)A common ingredient in cola drinks is phosphoric
acid, the same ingredient found in many rust removers. In terms of
hydrogen ion concentration, how much more acidic is cola, with a pH
of 2.5, than milk, with a pH of 6.6? Round your answer to the
nearest tenth. Since we are comparing the cola to the milk, set up
a ratio that has the cola concentration on the top and the milk
concentration on the bottom. MilkCola Compare cola to milk: Cola is
12589.3 times more acidic than milk.
Slide 16
Example 7 d) In terms of hydrogen ion concentration, how many
times more acidic is seawater than a soapy water solution? Since we
are comparing the sea water to the soapy water, set up a ratio that
has the sea water concentration on the top and the soapy water
concentration on the bottom. Soapy waterSea water Compare cola to
milk: Sea water is 10 000 times more acidic than soapy water.
Slide 17
Need to Know
Slide 18
Slide 19
To verify your value of a logarithm, substitute your answer
into the original equation,. You can estimate the value of a
logarithm of the form by converting x to a power with base b that
has approximately the same value as x, then solving for y.
Slide 20
Need to Know Many real-life situations have values that vary
greatly. A logarithmic scale with powers of 10 can be used to make
comparisons between large and small values more manageable. Three
examples of logarithmic scales are: Richter Scale used to measure
the magnitude of an earthquake pH Scale used to measure acidity of
a solution Decibel Scale used to measure sound level Youre ready!
Try the homework from this section.