Solving Exponential Equations
Summary
TLDRThis script offers a detailed tutorial on solving exponential equations without using logarithms. It covers various examples, starting with converting bases to simplify equations, such as changing base nine to base three. The tutorial demonstrates how to equate exponents once bases match and solve for variables like x. It further explores more complex scenarios, including dealing with squared terms and multiple bases, and concludes with a direct approach to equations that can't be simplified through base conversion, necessitating the use of logarithms for exact solutions.
Takeaways
- 🔢 Changing the base of an exponential equation can simplify solving for x, like converting 9 to 3^2.
- ➡️ Once bases are the same on both sides of the equation, set the exponents equal to each other.
- 🧮 Solving exponential equations often involves simplifying both sides to the same base, then isolating x.
- 📝 For more complex cases, like non-integer bases or when base changes aren't possible, logarithms can be used.
- ✔️ Use the properties of exponents, such as multiplication and power rules, to simplify exponential expressions.
- 💡 Taking the log or natural log of both sides helps solve exponential equations with different bases.
- ✖️ Quadratic factoring techniques can also be used when the equation is quadratic in form.
- 📏 In equations where a common base cannot be found, logarithms or natural logs offer a useful solution.
- 🔁 Exponent rules allow for conversion between forms like 27 = 3^3 or 81 = 3^4 to make solving easier.
- ✏️ The final solutions are often verified by back-substitution or using a calculator for approximate values.
Q & A
How do you solve the equation 3^(x+2) = 9^(2x-3) without using logarithms?
-You can change base 9 to base 3 since 3^2 = 9. Then, replace 9 with 3^2 and multiply the exponents: 2x - 3 becomes 2*(2x - 3) = 4x - 6. Now, with the same base, equate the exponents: x + 2 = 4x - 6. Solve for x to get x = 8/3.
What is the common base to convert 8 and 16 for the equation 8^(4x-12) = 16^(5x-3)?
-The common base is 2, since both 8 and 16 are powers of 2. Replace 8 with 2^3 and 16 with 2^4, then multiply the exponents accordingly.
How do you handle the equation 27^(3x-2) = 81^(2x+7)?
-Since 27 is 3^3 and 81 is 3^4, replace them with their base 3 equivalents. Then, equate the exponents: 3*(3x-2) = 4*(2x+7). Solve for x to find the value.
What is the value of x in the equation 3^x = 8?
-You can use logarithms to solve this. Taking the log of both sides gives x * log(3) = log(8). Then, x = log(8) / log(3), which is approximately 1.8928.
How do you find the value of x in the equation e^x = 7?
-Take the natural log of both sides to get x * ln(e) = ln(7). Since ln(e) is 1, x = ln(7), which is approximately 1.9459.
What steps are taken to solve the equation 5 + 4^(x-2) = 23?
-Subtract 5 from both sides to get 4^(x-2) = 18. Then, take the log of both sides and solve for x - 2. Finally, add 2 to both sides to find x.
How do you approach the equation 3 + 2e^(3-x) = 7?
-Subtract 3 from both sides to get 2e^(3-x) = 4. Divide by 2 to isolate e^(3-x) = 2. Take the natural log of both sides and solve for x.
What is the method to solve the equation 3^(x^2+4) = 1/27?
-Since 1/27 is 3^-3, make the bases the same and equate the exponents: x^2 + 4 = -3. Solve for x to find the values.
How do you handle the equation 2^(x^2) * 2^(3x) = 16?
-Since 16 is 2^4, replace it with 2^4. Add the exponents since it's multiplication with the same base: x^2 + 3x = 4. Solve the quadratic equation to find x.
What is the approach to solve the equation 4^(2x) - 20 * 4^(x) + 64 = 0?
-Let a = 4^x, then the equation becomes a^2 - 20a + 64 = 0. Factor the quadratic equation to find the values of a, and subsequently x.
Outlines
📚 Solving Exponential Equations Without Logs
This paragraph introduces a method for solving exponential equations without using logarithms. The first example involves transforming an equation with bases of 3 and 9 into a common base of 3, resulting in an equation where the exponents can be set equal to each other to solve for x. The process includes changing the base, multiplying exponents, and simplifying the equation to find x equals eight over three. The second example demonstrates converting bases of 8 and 16 into base 2, resulting in an equation where exponents are equated and solved to find x equals negative three. The third example shows converting bases of 27 and 81 into base 3, leading to an equation where exponents are equated and solved to find x equals thirty-four.
🔍 Solving Exponential Equations Using Logarithms
This section discusses solving exponential equations using logarithms. The first example shows how to solve for x in an equation where 3 raised to the power of x equals 8 by taking the logarithm of both sides, resulting in x being the log of 8 divided by the log of 3. The second example involves solving for x in an equation with base e, where e raised to the power of x equals 7, using the natural logarithm to find x equals the natural log of 7. The third example deals with an equation where 4 raised to the power of x minus 2 equals 23, and after subtracting 5 from both sides, the logarithm is taken to solve for x, resulting in x equals 2 plus the log of 18 divided by the log of 4. The final example in this paragraph solves for x in an equation involving e raised to the power of 3 minus x, leading to x equals 3 minus the natural log of 2.
🧩 Factoring and Solving Exponential Equations
This paragraph focuses on factoring techniques to solve exponential equations. The first example involves an equation where 3 raised to the power of x squared plus 4 equals one over 27, and by recognizing that 3 to the power of -3 equals 1/27, the equation is simplified to x squared plus 4x equals -3, which factors to (x+3)(x+1), yielding solutions of x equals -3 and x equals -1. The second example shows multiplying two exponential expressions with the same base, leading to an equation that simplifies to x squared plus 3x equals 4, which factors to (x+4)(x-1), resulting in x equals -4 and x equals 1. The third example involves an equation that simplifies to a quadratic equation, where 4 raised to the power of 2x minus 20 equals 4 raised to the power of x plus 64, leading to solutions of x equals 1 and x equals 2. The final example in this paragraph involves an equation that simplifies to 3 to the 2x equals 27, resulting in x equals 3/2.
🔢 Solving Complex Exponential Equations
This final paragraph presents a complex exponential equation where 3 to the power of 2x minus 3 to the power of 2x minus 1 equals 18. The solution involves dividing both sides by 3 to the power of -1, resulting in a complex fraction that simplifies to 3 to the power of 2x equals 27. Recognizing that 27 is 3 to the power of 3, the equation is further simplified to find that 2x equals 3, leading to the solution x equals 3/2. This example demonstrates the process of simplifying complex exponential equations through division and recognizing equivalent powers.
Mindmap
Keywords
💡Exponential equation
💡Changing bases
💡Exponent multiplication
💡Setting exponents equal
💡Factoring
💡Quadratic equation
💡Taking logarithms
💡Natural logarithm
💡Exponent properties
💡Complex fraction simplification
Highlights
Transforming base nine into base three simplifies the exponential equation.
Exponents can be multiplied when raising to another exponent.
Equating exponents when bases are the same is a key step in solving.
Solving for x involves subtracting and adding to isolate the variable.
Finding x as a fraction by simplifying the equation.
Using a common base, base two, to convert 8 and 16 in another example.
Multiplying exponents by the powers of the common base.
Setting exponents equal to each other after converting to the same base.
Solving for x by isolating the variable through subtraction and addition.
Using base three to solve an equation involving 27 and 81.
Replacing numbers with their base three equivalents simplifies the equation.
Solving for x by setting exponents equal after base conversion.
Using logarithms to solve an equation when direct methods are not applicable.
Demonstrating the process of taking logarithms to isolate x.
Using natural log to solve an equation involving base e.
Solving an equation with a non-exponential term by subtracting and taking logs.
Converting an equation to a quadratic form by factoring and substitution.
Solving a complex exponential equation by factoring out the greatest common factor.
Isolating x by dividing both sides of the equation by a common base.
Transcripts
consider the following exponential
equation
three raised to the x plus two
is equal to nine raised to the two x
minus three
how can we find the value of x without
using logs
what you need to do is change
base nine into base three
three squared is equal to nine
so we can replace nine
with three squared
and whenever you raise one exponent to
another exponent you need to multiply so
we got to multiply two by two x minus
three
so on the right side this is going to be
three raised to the four x minus six
now
if the bases are the same
then the exponents
must be equal to each other
therefore
x plus 2
is equal to four x
minus six
now let's subtract both sides by x
and let's add six to both sides
so these two will cancel
two plus six is eight
four x minus x is three x
so we can see that x
is equal to eight over three
and so that's the answer for this one
now let's work on another example
8
raised to the 4x minus 12
is equal to 16
raised to the 5x minus 3.
so go ahead and find the value of x now
what we need to do is convert 8 and 16
into a common base
2 is a multiple of 8 and 16 so 2 is the
common base
2 to the third power is 8
2 to the fourth power 16.
so let's replace 8
with 2 to the third power
and let's replace 16
with 2 to the fourth power
now we need to multiply 4x minus 12 by
3.
4x times 3 is 12x
3 times negative 12 is negative 36.
now we need to multiply 5x minus 3 by 4.
so that's going to give us 20x
minus 12.
so we could set these two
equal to each other now that we have the
same base
so 12x minus 36
is equal to 20x minus 12.
let's subtract both sides by 12x
let's add
12 to both sides
so negative 24
is equal to 8x
therefore x
is equal to negative three
let's try one more example
27
raised to the 3x minus 2.
let's say that's equal to 81
raised to the two x plus seven
so try this problem
now three to the third power is equal to
twenty seven
and three to the fourth power is equal
to eighty one
so let's replace twenty seven with three
cubed
and let's replace eighty-one
with three to the fourth power
three times three x minus two
that's going to be nine x minus six
and four times two x plus seven
that's eight x plus twenty eight
so now that we have the same base
we can set the exponents
equal to each other
so nine x minus six
is equal to eight x plus twenty eight
so let's subtract both sides by eight x
and let's add
six to both sides
so these will cancel
nine x minus eight x is x
twenty eight plus six is thirty four
so x
is equal to thirty four in this example
three raised to the x is equal to eight
what is the value of x
in order to find it we can take the log
or the natural log of both sides it
really doesn't matter which one you're
going to use
let's take the log of both sides
once you do that you can move the
exponent to the front
so x log 3
is equal to log 8.
so x is log eight
divided by log three
if you type this in
you should get
1.8928
and you can check your answer what is 3
raised to the 1.8928
if you type that in
the calculator will give you this answer
eight point zero zero
zero zero nine
granted this is rounded so this is going
to be very close to eight
what about this one if e raised to the x
is equal to 7
what is the value of x
now whenever you're dealing with base e
it's better to use the natural log as
opposed to the regular log
so we're going to take the natural log
of both sides
and we're going to move x to the front
so x ln e
is equal to ln7
and
e is one
so x times one is x
so the answer is l and seven
this is the exact answer and then you
could type it in if you want the decimal
answer
so the decimal answer is 1.9459
so that's the approximation
now let's work on this example 5 plus 4
raised to the x minus 2
is equal to 23
let's find the value of x
so let's begin by subtracting both sides
by 5.
we can't take the log of both sides yet
it wouldn't be wise
23 minus 5 is 18.
now at this point we can't really
convert
18 into a base two
two to the fourth is 16 2 to the fifth
is 32
so
we can't really change 18 to a base two
so we have to take the log of both sides
you can also use the natural log too
it will work as well
now we can move the exponent
to the front
and so it's x minus 2 in parentheses
times log 4
and that's equal to log 18.
i wouldn't recommend distributing
log 4 to x minus two
instead
it's better to divide both sides by log
four
so x minus two
is equal to
log 18.
well let's get the exact answer first so
this is equal to log 18 over log 4.
the exact answer is this x is equal to 2
plus
log 18 over log 4.
log 18
divided by log 4
that's about 2.08496
and if we add 2 to it
we can see that x
is equal to 4.08496
so that's the answer
let's try this problem
3 plus 2
e raised to the 3 minus x
let's say it's equal to 7.
go ahead and find the value of x
so before we take the natural log of
both sides let's subtract both sides by
three
so two e to the three minus x
is equal to four
next let's divide both sides by two so e
to the three minus x is equal to two
and now we can take the natural log of
both sides
now let's move the exponent to the front
so three minus x times ln e
is equal to ln2 and lne is one so
this is just gonna be three minus x
now what i'm gonna do is take the
negative x move it to this side
it's negative x on
the left side but it's gonna be positive
x
on the right side
now i'm going to take ln 2 move it to
this side
which is going to change from positive
to negative
so x
is 3 minus ln 2.
that's the exact answer the decimal
value
is 2.3069
how would you solve
this equation
three
raised to the x squared plus four
let's say that's equal to one over
twenty seven
what would you do in order to find the
value of x
in this case
we know that three to the third is equal
to twenty seven
so three to the negative three is one
over twenty seven
we need to make the bases the same
so this is 3 to the negative 3.
because the bases are the same
we can now make the exponents
equal to each other which means that x
squared plus 4x
is equal to negative 3.
now let's move the negative three to the
left side
in which case is going to be positive
three and we could factor two numbers
that multiply to three but add to four
are three and one
so this is x plus three
and x plus one
so therefore x
can be equal to negative three
and x is equal to negative one
here's another example
two raised to the x squared
times two raised to the three x
is equal to sixteen
find the value of x
two to the fourth is equal to sixteen
so let's replace sixteen with that
now let's say if you're multiplying by a
common base
you can add the exponents 2 plus 3 is 5.
in this case 2 is the common base
so we can add x squared and three x
so this is two x squared plus three x
so now we can set x squared plus three x
equal to four
now let's subtract both sides by four
so x squared plus three x minus four
is equal to zero and let's factor
two numbers that multiply to negative
four but add to positive three
are positive four and negative one
so it's x plus four
times x minus one which means that x
is equal to negative four
and it's equal to positive one
try this
4 raised to the 2x
minus 20
multiplied by 4
raised to the x
plus 64
is equal to 0.
so what can we do
in order to find the value of x
what i think we should do
is factor
but factor by substitution
this equation can be reduced to a
quadratic equation
we're going to set a equal to 4 raised
to the x
which means that a squared is 4 to the
2x
so let's replace 4 to the 2x with a
squared
and let's replace 4 to the x with a
what two numbers multiply to 64 but add
to negative 20.
this is negative 16 and negative 4.
so to factor it it's a minus 16
and a minus 4.
so therefore a
is equal to 16 and a is equal to 4.
and we know that a is 4 to the x
so therefore 4 raised to the x is equal
to 16
and 4 to the x is equal to 4.
4 is basically 4 to the first power
so we can clearly see that x
is equal to 1.
now 16
is equivalent to 4 squared
so 4 to the x is equal to 4 squared
which means that x
is equal to 2
and so those are the two answers
let's try one more example
three to the two x
minus three
to the two x minus one
let's say it's equal to eighteen
go ahead and find the value of x
so we can't really factor in this
example
because 2x is not twice the value of 2x
minus 1.
but we could factor the gcf
we could take out
a 3 to the 2x
3 to the two x divided by itself
is equal to one
now
three to the two x minus one divided by
three to the two x
if we uh write it out
notice what we'll get
let's ignore the negative sign for now
we know the overall answer is going to
be negative
when you divide by a common base
you need to subtract the exponents
2x minus 1 minus 2x
the 2x will cancel and so it's going to
be 3 to the negative 1.
and so that's what we have right now
and if you distribute
you can see that you're going to get the
original equation
3 to the 2x times 1 gives you this term
and 3 to the 2x times 3 to negative 1
you add the exponents 2x plus negative 1
is 2x minus 1.
now let's divide both sides
by this
three to the negative one is basically
one third
so we have a complex fraction let's
multiply the top and the bottom by three
so eighteen times three is fifty four
three times one is three
three times a third is one
and
three minus 1 is 2 54 over 2 is 27
so 3 to the 2x
is equal to 27
and 27
is 3 to the third power
so 2x is equal to 3
which means that x
is 3 over 2
and so that's the answer
you
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