Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation z^3=-i
-
Equation 3*x^2+x-10=0
-
Equation 2x=5
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Equation 5^(9-x)=64
- Express {x} in terms of y where:
- -8*x+14*y=5
- 17*x+7*y=-6
- -11*x-14*y=6
- -18*x-8*y=-5
- Sum of series:
-
64
-
Identical expressions
- five ^(nine -x)= sixty-four
- 5 to the power of (9 minus x) equally 64
- five to the power of (nine minus x) equally sixty minus four
- 5(9-x)=64
- 59-x=64
- 5^9-x=64
-
Similar expressions
- 5^(9+x)=64
5^(9-x)=64 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$5^{9 - x} = 64$$
or
$$5^{9 - x} - 64 = 0$$
or
$$1953125 \cdot 5^{- x} = 64$$
or
$$\left(\frac{1}{5}\right)^{x} = \frac{64}{1953125}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{5}\right)^{x}$$
we get
$$v - \frac{64}{1953125} = 0$$
or
$$v - \frac{64}{1953125} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{64}{1953125}$$
We get the answer: v = 64/1953125
do backward replacement
$$\left(\frac{1}{5}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(\frac{64}{1953125} \right)}}{\log{\left(\frac{1}{5} \right)}} = - \frac{6 \log{\left(2 \right)}}{\log{\left(5 \right)}} + 9$$
$$5^{9 - x} = 64$$
or
$$5^{9 - x} - 64 = 0$$
or
$$1953125 \cdot 5^{- x} = 64$$
or
$$\left(\frac{1}{5}\right)^{x} = \frac{64}{1953125}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{5}\right)^{x}$$
we get
$$v - \frac{64}{1953125} = 0$$
or
$$v - \frac{64}{1953125} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{64}{1953125}$$
We get the answer: v = 64/1953125
do backward replacement
$$\left(\frac{1}{5}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(\frac{64}{1953125} \right)}}{\log{\left(\frac{1}{5} \right)}} = - \frac{6 \log{\left(2 \right)}}{\log{\left(5 \right)}} + 9$$
Sum and product of roots
[src]
sum
/ 3 \ | ------| | log(5)| log\125/4 /
$$\log{\left(\left(\frac{125}{4}\right)^{\frac{3}{\log{\left(5 \right)}}} \right)}$$
=
/ 3 \ | ------| | log(5)| log\125/4 /
$$\log{\left(\left(\frac{125}{4}\right)^{\frac{3}{\log{\left(5 \right)}}} \right)}$$
product
/ 3 \ | ------| | log(5)| log\125/4 /
$$\log{\left(\left(\frac{125}{4}\right)^{\frac{3}{\log{\left(5 \right)}}} \right)}$$
=
6*log(2)
9 - --------
log(5)
$$- \frac{6 \log{\left(2 \right)}}{\log{\left(5 \right)}} + 9$$
9 - 6*log(2)/log(5)
Rapid solution
[src]
/ 3 \
| ------|
| log(5)|
x1 = log\125/4 /
$$x_{1} = \log{\left(\left(\frac{125}{4}\right)^{\frac{3}{\log{\left(5 \right)}}} \right)}$$
x1 = log((125/4)^(3/log(5)))
The graph