Web & Social Media Analytics Previous Year Question Paper.pdf
8.4 logarithmic functions
1.
2. What is a Logarithm?
We know 22 = 4 and 23 = 8, but for what
value of x does 2x = 6?
It must be between 2 and 3…
Logarithms were invented to solve
exponential equations like this.
x = log26 ≈ 2.585
3. Logarithms with Base b
Let b and y be positive numbers and b≠1.
The logarithm of y with base b is written
logby and is defined:
logby = x if and only if bx = y
8. Common and Natural Logs
Common Logarithm - the log with base 10
Written “log10” or just “log”
log10 x = log x
Natural Logarithm – the log with base e
Can write “loge“ but we usually use “ln”
loge x = ln x
9. Evaluating Common and
Natural Logs
Use “LOG” or “LN” key on calculator.
Evaluate. Round to 3 decimal places.
log 5
ln 0.1
10. Evaluating Log Functions
The slope s of a beach is related to the
average diameter d (in mm) of the sand
particles on the beach by this equation:
s = 0.159 + 0.118 log d
Find the slope of a beach if the average
diameter of the sand particles is 0.25 mm.
11. Inverses
The logarithmic function g(x) = logb x
is the inverse of the exponential function
f(x) = bx.
Therefore:
g(f(x)) = logb bx = x and f(g(x)) = blogb x = x
This means they “undo” each other.
14. Finding Inverses
Switch x and y, then solve for y.
Remember: to “chop off a log” use the
“circle cycle”!
Find the inverse:
y = log3 x y = ln(x + 1)
16. Logarithmic Graphs
Remember f-1 is a reflection of f over the
line y = x.
Logs and exponentials are inverses!
exp. growth exp. decay
17. Properties of Log Graphs
General form: y = logb (x – h) + k
Vertical asymptote at x = h.
(x = 0 for parent graph)
Domain: x > h
Range: All real #s
If b > 1, graph moves up to the right
If 0 < b < 1, graph moves down to the
right.
18. To graph:
Sketch parent graph (if needed).
Always goes through (1, 0) and (b, 1)
Choose one more point if needed.
Don’t cross the y-axis!
Shift using h and k.
Be Careful: h is in () with the x, k is not