h-Math 30 - Exponential Functions to Logarithmic Functions.pdf ...
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h-Math 30 - Exponential Functions to Logarithmic Functions.pdf ...
Sign in. Page. 1. /. 6. Loading⦠Page 1 of 6. Exponential Functions to Logarithmic. Functions: Math 30 Pure. Consider
Exponential Functions to Logarithmic Functions: Math 30 Pure
Consider the function below: y
2(3) x
4
Domain: x R Range: y > 4, y
R
Determine the equation of the inverse of y
2(3) x
4.
Step 1: Swap “ x ” and “ y ” x
2(3) y
4
Step 2: Isolate “ y ” x 4 2(3) y 1 ( x 4) 3 y 2
Range: y R Domain: x > 4, x R
Solve for the x-intercept algebraically. Set y
0 in the equation y
0 log3 1 (x 2 1 (x 2 x 4 x 4 x 6
1 ( x 4) 2
log 3
1 ( x 4) 2
and solve for “ x ”.
convert to exponential form…
4) 30 4) 1
2 2
The x-intercept in this example is 6, which corresponds to the y-intercept in the original function y 2(3) x 4 .
Consider another example:
Solve for y-intercept: y
y
y
3 0 (2) 6 2 3 6 2 4.5
Solve for x-intercept: 3 x (2) 6 2 3 x 6 (2) 2 4 2x x 2 0
Determine the equation of the inverse: y
3 x (2) 6 x 2
3 y (2) 6 x 6 2
Solve for x-intercept: 0 log 2 20
1 3 2
2 ( x 6) 3
2 ( x 6) 3
2 ( x 6) 3 x 6
x
6
x
4.5
3 2
3 y 2 (2) ( x 6) 2 3
2y y
Solve for y-intercept: 2 (6) 3
y
log 2
y
log 2 4
y
2
log 2
2 ( x 6) 3
Conclusions:
1 ( x 10) 7 vertical asymptote: x 10 domain: x > 10, x R
Given the function y
log 2
The x-intercept should be 7 units to the right of the vertical asymptote. This is referred to as a horizontal stretch about the line x 10 when being compared to the graph defined by the function y log 2 ( x 10) . Proof: Solve for the x-intercepts for the following:
log 2
1 ( x 10) 7
0 log 2 ( x 10)
0 log 2
1 ( x 10) 7
20
20
y
y
log 2 ( x 10)
x 10
1 ( x 10) 7 7 x 10
1
1 x 10
x
1 ( x 10) 7
9 x
3
One more example to consider:
y
log 2
1 ( x 3) 5
vertical asymptote: x 3
From what we’ve discovered so far, the x-intercept should be 5 units to the left of x 3 .
