Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x/4+x=4
-
Equation x^2+11=0
- Equation x+y=5
-
Equation 1*(x-1)*(x-3)=0
- Express {x} in terms of y where:
- -11*x-3*y=14
- -1*x-17*y=-12
- 20*x+18*y=-9
- -2*x+20*y=-2
- Limit of the function:
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(3/2)^x
-
Identical expressions
- (three / two)^x= sixteen / eighty-one
- (3 divide by 2) to the power of x equally 16 divide by 81
- (three divide by two) to the power of x equally sixteen divide by eighty minus one
- (3/2)x=16/81
- 3/2x=16/81
- 3/2^x=16/81
- (3 divide by 2)^x=16 divide by 81
(3/2)^x=16/81 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(\frac{3}{2}\right)^{x} = \frac{16}{81}$$
or
$$\left(\frac{3}{2}\right)^{x} - \frac{16}{81} = 0$$
or
$$\left(\frac{3}{2}\right)^{x} = \frac{16}{81}$$
or
$$\left(\frac{3}{2}\right)^{x} = \frac{16}{81}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{3}{2}\right)^{x}$$
we get
$$v - \frac{16}{81} = 0$$
or
$$v - \frac{16}{81} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{16}{81}$$
We get the answer: v = 16/81
do backward replacement
$$\left(\frac{3}{2}\right)^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(\frac{3}{2} \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(\frac{16}{81} \right)}}{\log{\left(\frac{3}{2} \right)}} = -4$$
$$\left(\frac{3}{2}\right)^{x} = \frac{16}{81}$$
or
$$\left(\frac{3}{2}\right)^{x} - \frac{16}{81} = 0$$
or
$$\left(\frac{3}{2}\right)^{x} = \frac{16}{81}$$
or
$$\left(\frac{3}{2}\right)^{x} = \frac{16}{81}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{3}{2}\right)^{x}$$
we get
$$v - \frac{16}{81} = 0$$
or
$$v - \frac{16}{81} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{16}{81}$$
We get the answer: v = 16/81
do backward replacement
$$\left(\frac{3}{2}\right)^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(\frac{3}{2} \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(\frac{16}{81} \right)}}{\log{\left(\frac{3}{2} \right)}} = -4$$
Sum and product of roots
[src]
sum
0 - 4
$$-4 + 0$$
=
-4
$$-4$$
product
1*-4
$$1 \left(-4\right)$$
=
-4
$$-4$$
-4
The graph