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Introduction
An exponential equation is one
that has exponential expressions,
in other words, powers that have in
their exponent expressions with the
unknown factor x.
In this section, we will resolve the
exponential equations without using logarithms.
This method of resolution consists in reaching an equality of the
exponentials with the same base in order to equal the exponents.
For example:
$$ 3^{2x} = 3^6 $$
Obviously, the value that x has to take for the equality to be true is 3.
In order to achieve this type of expressions we have to factorize,
express all the numbers in the form of powers,
apply the properties of powers and write roots as powers.
Sometimes we will need to make a change of variable to transform
the equation in a quadratic one.
We can also resolve using logarithms, but we will leave this type of
procedures for more difficult equations with different bases in the
exponential expressions, making it impossible to use the previous
method of equalizing.
For example,
$$ 3^{x+3} = 5^x $$
which has a real solution, using logarithms of,
$$ x = \frac{3 ln(3)}{ln\left(\frac{5}{3}\right)} $$
Before we start...let's remember the properties of powers
Product

Power

Quotient

Negative exponent

Inverse

Inverse of inverse

Solved Exponential Equations
Equation 1
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Taking into account that
$$ 27=3^2 $$
We can rewrite the equation as
Therefore,
Equation 2
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Taking into account that
We can rewrite the equation as
Therefore,
Equation 3
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Taking into account that
We operate the expression using the properties of powers
Therefore, we have the linear equation
Equation 4
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Taking into account that
We can rewrite the equation as
This way, we can extract a common factor of 2^{ x}:
Therefore,
Equation 5
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Taking into account that
We can rewrite the equation as
We have the common base 3^{ x}, but because one of them is squared,
we write
Substituting, the equation finishes like
In other words, a quadratic equation:
We multiply the full equation by 9:
We solve it:
Therefore,
So, we obtain
The second option is not possible because it is negative. Therefore,
From where we obtain
Equation 6
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Taking into account that
We rewrite the equation as
We call t = 3^{ x}:
Substituting, the equation finishes like
We resolve the quadratic equation:
So,
So,
The second solution is not possible because it is negative, but the first one is.
Equation 7
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Taking into account that
We rewrite the equation as
We call
Substituting, the equation finishes like
We resolve the previous equation:
Therefore
The second option is not possible. So,
Therefore,
Equation 8
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Taking into account that
We rewrite the equation as
We call
Substituting, the equation finishes like
We resolve the quadratic equation:
Therefore
Notice that
So, both are powers of 3.
Then the two solutions are
Equation 9
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Taking into account that
We can rewrite the equation as
We call
Substituting, the equation finishes like
We resolve the equation:
So we have
But the solutions
are not possible because one is zero and the other is negative.
So, the only one solution is
Equation 10
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Taking into account that
We can rewrite the equation as
Therefore,
Equation 11
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Taking into account that
We can rewrite the equation as
We call
Substituting, we obtain a quadratic equation
We resolve it:
So
The first solution is not possible because it is zero. Then,
Equation 12
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Taking into account that
we rewrite the equation as
We call
Substituting, we obtain a quadratic equation:
We resolve it:
Therefore,
The first solution is not possible because it is zero.
Therefore, the solution is x = 1.
Equation 13
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Notice that
So, the equation can be written as
As we have an exponential dividing,
we multiply the full equation by it and that way it disappears:
Now we call
Substituting, the equation finishes like
We resolve it:
Therefore,
The second solution is not possible because it is negative.
So, the solution is
Equation 14
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Notice that
So we write the equation as
We call
So,
Substituting, we obtain a quartic equation (equation of degree four):
We resolve it:
The first solution is not possible because it is zero.
So,
Equation 15
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Note that
So we can write the equation as
As we have an exponential dividing,
we multiply the full equation by it and that way it disappears:
We call
Substituting, we have a cubic equation (degree three)
We rewrite it
We apply Ruffini's Rule:
One of the solutions is t = 4.
We calculate the other two:
But these are not possible solutions because they are negative.
Therefore,
Equation 16
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We rewrite the equation:
Therefore, the solution is
Equation 17
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We rewrite the equation:
We multiply the full equation by the exponential and that way it disappears:
Therefore,
Equation 18
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Note that
We rewrite the equation:
We operate:
Therefore,
Equation 19
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Note that
So we rewrite the equation as
We call
Then we obtain the expression
Now we define
Notice that
We are going to suppose that
Therefore
So
And this is not possible.
So, let's suppose
Therefore,
One solution would be t = 0 but, like before, is not possible.
The other solution is
But we have supposed that k = 1 and we have to check that it is true:
Because it's true, the solution to the equation is x = 2.
Equation 20
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We rewrite the equation as
We use 25 = 5^{2} and the properties
of powers (we write the root in the form of a power):
The, we have
Finally, we resolve the equation:
Equation 21
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We write the roots in the form of powers
We want for this to be true:
Therefore, we have two solutions:
$$ x = 0,\ x = 2 $$
Equation 22
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We have three nested roots (one inside the other).
We write the roots in the form of powers.
The equations ends up like
We resolve the equation and we obtain the solution
Equation 23
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We write the roots in the form of powers:
We want for this to be true:
Finally, we resolve the linear equation:
Equation 24
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We write the root in the form of powers:
We want this to be true:
Taking into account that x cannot be 0.
We apply Ruffini's Rule:
One solution is x = 2. We calculate the others:
There aren't any real solutions.
Therefore, the only solution to the exponential equation is
$$ x = 2 $$
Equation 25
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We write the roots in the form of powers.
The equation ends up like:
Note that 8 = (2)^{3}
We want this to be true:
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