MATH 120
Week 9
Properties of Logarithmic Functions
NAME
Logarithmic Functions
Since it has been a few weeks since we talked about logarithmic functions, let’s review a couple of things...
Let b denote an arbitrary positive constant other than 1.
The exponential function with base b is defined by the equation
y = bx .
Our logarithmic functions are defined to be the inverses of the exponential functions.
We define the expression logb x to mean ’the exponent to which b must be raised to yield x’.
In other words, log2 8 says...
Since exponential and logarithmic functions are related, this means we can express logb x = y in terms of
exponents.
Now, just like any algebraic operation has a variety of properties (i.e. addition is commutative) the logarithmic function has a variety of properties.
Some of these properties I will have you derive on your own. Others you will be expected to be able to pove
(i.e. explain why).
Hint: When asked to explain ’why’, try writing the logarithim in exponential form. Write logb P and logb Q
in exponential form.
Properties of Logarithmic Functions
Assume that P and Q are positive real numbers.
1. (a) logb b =
(b) logb 1 =
1
2. logb P Q = logb P + logb Q
Why?
3. logb (P/Q) = logb P − logb Q
Why?
4. logb P n = n logb P
Why?
5. blogb P =
6. logb bx =
We can use these properties to rewrite logarithmic expressions in a variety of different ways.
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Examples
Write each expression as a single logarithm with a coefficient of 1 (or simplify completely where possible).
1. log10 50 − log10 5
2. log5 6 + log5 (1/3) + log5 10
3. p logb A − q logb B + r logb C
We can also use these properties in order solve equations and inequalities.
Solve the following equations and inequalities.
1. 3e1+t = 9
2. 4x = 32x+1
3
3. log3 x + log3 (x + 2) = 1
Using the properties of logarithms to solve logarithmic equations may introduce extraneous
solutions that do not work in the original equation.
(This happens because the logarithm function requires positive inputs but in solving an
equation, we know ahead of the time the sign of an input involving a variable.)
*Be sure to check your solutions.*
4. ln(2 − 3x) ≤ 1
5. e2+x ≥ 100
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Change of Base
Sometimes, you may be given an expressions with logarithms of multiple different bases. This (or other
concerns) may cause you to want to change the base of the logarithm.
Change of Base Formula
loga x =
Examples
Express each quantity in terms of natural logarithms.
1. log2 3
2. log10 e
Some Things to Note About Logarithms
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logb x
logb a