"log" is short for "logarithm" .
This can probably explain better than I can: https://youtu.be/Z5myJ8dg_rM
Here's a simpler example of finding the inverse function:
f(x) = x + 1
Instead of f(x) , let's say y .
y = x + 1 Solve for x .
y - 1 = x
Swap x and y .
x - 1 = y This is the inverse.
f-1(x) = x - 1
In this context g-1(x) does not mean g(x) raised to the power of negative 1 .
It means the inverse function of g(x) . We can find g-1(x) like this....
g(x) = 9^x - 3
Instead of g(x) , write y .
y = 9^x - 3
Now we want to solve this equation for x .
y + 3 = 9^x
log(y + 3) = log(9^x)
log(y + 3) = x log 9
log(y + 3) / log 9 = x
To get the inverse function, swap x and y .
log(x + 3) / log 9 = y This is the inverse function. We can say...
g-1(x) = log(x + 3) / log 9
So... g-1(x) does not mean 1 / g(x) .
Neither does sin-1x = (sin x)-1 , even though sin2x = (sin x)2
This is something that has bothered me before. Maybe there is a reason why it is okay and I just don't understand it......?
distance between warehouse and Abby = distance between warehouse and Cardenas
Let (x, y) be the coordinates of the warehouse.
distance between (x, y) and (-3, -2) = distance between (x, y) and (3, 6)
√[ (x + 3)2 + (y + 2)2 ] = √[ (x - 3)2 + (y - 6)2 ]
(x + 3)2 + (y + 2)2 = (x - 3)2 + (y - 6)2
x2 + 6x + 9 + y2 + 4y + 4 = x2 - 6x + 9 + y2 - 12y + 36
6x + 4y + 4 = -6x - 12y + 36
16y = -12x + 32
y = -3/4x + 2 The warehouse can be any point along this line.
Here's a graph: https://www.desmos.com/calculator/eenwtk2dms
The warehouse needs to be on the line y = -3/4x + 2 and needs to be 4 units away from the road. So there are two possible locations.
I think the answer to this part is 6.4 miles, but I don't have a good explanation.
Remember, similar triangles can have different side lengths.
In order for two triangles to be similar, the angles must be the same.
Since ∠DCE and ∠BCA are vertical angles, they have the same measure.
Since ∠CDE and ∠CBA are right angles, they have the same measure.
And since two of the angles are the same, the third angle must be the same,
so △ABC is similar to △EDC by the Angle-Angle similarity rule.
To solve this, we have to know that △ABC is similar to △EDC .
Because △ABC is similar to △EDC , we can say that
22/100 = 32/x , where x is the length of AB in feet.
x = 3200/22 ≈ 145.45
So the width of the river is about 145.45 feet, just as you found!
Remember, the Pythagorean theorem says...
(leg)2 + (other leg)2 = (hypotenuse)2
Since the legs of this triangle are sides of a square, they must be the same length.
Let's call the length of each leg s , and plug in 5√2 for the hypotenuse.
s2 + s2 = ( 5√2 )2
Now we can solve this equation for s .
2s2 = ( 5√2 )2
2s2 = 5√2 * 5√2
Multiplication can be done in any order.
2s2 = 5 * 5 * √2 * √2
And √2 * √2 = 2
2s2 = 25 * 2
Divide both sides of the equation by 2 .
s2 = 25
Take the positive, since s is side length, square root of both sides.
s = 5