Pre – Calculus Math 12
Understanding Logarithms
Lesson Focus: To demonstrate that a logarithmic function is the inverse of an exponential function; to sketch
the graph of y = logc x, c > 0, c ≠ 1; to determine the characteristics of the graph of y = logc x, c > 0, c ≠ 1; to
explain the relationship between logarithms and exponents; to express a logarithmic function as an exponential
function and vice versa.
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a logarithm is the exponent to which a fixed base must be raised to obtain a specific value
i.e. in 53  125 , the logarithm of 125 is the exponent that must be applied to base 5 in order to obtain 125
(in this case the logarithm is 3) therefore log5 125  3
equations in exponential form can be written in logarithmic form and vice versa
i.e. x  c y (exponential form) and y  log c x (logarithmic form)
the inverse of the exponential function y  c x , c  0, c  1 is x  c y or in logarithmic form y  log c x
conversely, the inverse of the logarithmic function y  logc x, c  0, c  1 is x  log c y or in exponential
form y  c x
the graphs of an exponential function and its inverse logarithmic function are reflections of each other in the
line y  x
for the logarithmic function y  logc x, c  0, c  1 , the:
1. domain is  x x  0, x  
2. range is  y y  
3. x-intercept is 1,0 
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4. vertical asymptote is x  0 (the y-axis)
a common logarithm has base 10
 it is not necessary to write the base for common logs  log10 x  log x 
e.g. The graph of y  2 x is shown below. State the inverse of the function (in both exponential and logarithmic
form). Sketch the graph of the inverse of the function. Identify the following characteristics of the graph:
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domain
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range
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x-intercept (if it exists)
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y-intercept (if it exist)
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equation of the asymptote
e.g. For each expression in exponential form, rewrite it in logarithmic form. For each expression in
logarithmic form, rewrite it in exponential form.
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2
1. 3  81
2. 36  6
3. log 4 64  3
4. log10000  4
1
5. 4 3  64
6. log 3 35
4
e.g. Evaluate each expression.
1. log 2 32
2. log100 10
3. log3 27
4. log 4 x  3
5. log x 8  3
6. log 0.01  x
e.g. The intensity of sound is measured in decibels (dB). The level of a sound, L, in decibels, is given by
I 
L  10log   , where I is the intensity of the sound and I0 is the faintest sound detectable to humans.
 I0 
The sound level inside a particular car is 39 dB when it is idling and 80 dB at full throttle.
a) Let Ii be the intensity of the sound at idle and If be the intensity at full throttle. Write an expression for
both situations.
b) Rewrite each expression in exponential form. Rearrange each expression to isolate Ii and If .
c) To compare the intensities, divide If by Ii. How many times more intense is the sound at full throttle as
compared to idle?