Download Tech Math 2 Lecture Notes, Section 13.2

Tech Math 2 Notes
Section 13.2
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Section 13.2: Logarithmic Functions
Big Idea: The logarithmic function “undoes” the exponential function.
If y = bx, then x = logb y.


The first formula is in “exponential form,” while the second equation is in “logarithmic form.”
The logarithm is just an exponent. In fact, it helps sometimes to not to say the word logarithm, but
instead “the exponent on b that gives y”
In the past, we have dealt with functions where we had a variable x raised to a constant power, and to “undo”
these equations and calculate what x is, we took roots:
y = x2
x y
5
1
y=x
x y 5
y = x0.9
xy
10
9
Now we are dealing with exponential functions, where x is the exponent, so to “undo” these equations and
calculate what x is, we take logarithms:
y = 2x
x = log2 y
y = 10x
x = log10 y = log y
y = ex
x = loge y = ln y
Facts about the logarithmic function for b > 1:
1. The constant number b is called the base.
2. Some common bases are 2, 10, and e.
3. The variable y ends up being any real number (i.e., the range of the function is all real numbers).
4. The variable x can only take on values greater than zero (i.e., the domain is x > 0).
5. The graph always passes through the point (1, 0), because any number raised to the power of zero equals
one.
6. The logarithm of a fraction is negative.
7. The logarithm of a number greater than one is positive.
8. The graph of the function increases, as shown below:
Tech Math 2 Notes
Section 13.2
Practice:
1. Write 23 = 8 in logarithmic form.
2. Write 4-2 = 1/16 in logarithmic form.
3. Write 641/3 = 4 in logarithmic form.
4. Write log2 32 = 5 in exponential form.
5. Write log3 (1/9) = -2 in exponential form.
6. Solve for x: log5 125 = x
7. Solve for N: log8 (N + 1) = 3
8. Solve for b: logb 625 = 4
9. Graph y = log3 x
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