Solving Exponential Equations

Summary

This 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

00:00

📚 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.

05:03

🔍 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.

10:03

🧩 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.

15:06

🔢 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.

Keywords

💡Exponential equation

An exponential equation is a mathematical equation where variables appear as exponents. In the video, the speaker solves various exponential equations, such as 'three raised to the x plus two equals nine raised to the two x minus three'. These equations are central to the video as they show how to manipulate bases and exponents to solve for unknown values.

💡Changing bases

Changing bases refers to rewriting numbers or terms with a different base, typically to simplify calculations. For instance, the speaker changes 'nine' to 'three squared' because working with the same base (base 3) simplifies the process of solving the equation. This technique is used throughout the video to make the exponents comparable.

💡Exponent multiplication

When one exponent is raised to another, their powers are multiplied. The video demonstrates this rule, such as when '3^2' is raised to '2x - 3', resulting in '3 raised to the 4x - 6'. This concept is crucial in handling more complex exponential terms in equations.

💡Setting exponents equal

In an equation where the bases on both sides are the same, the exponents can be set equal to each other. The speaker frequently uses this step to transition from exponential expressions to simpler linear or quadratic equations, such as 'x + 2 equals 4x - 6'.

💡Factoring

Factoring involves breaking down an expression into simpler terms that, when multiplied together, give the original expression. The video uses factoring to solve equations after exponents are set equal, like factoring 'x squared + 4x + 3'. This is used to solve for the variable 'x'.

💡Quadratic equation

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0. In the video, the speaker often derives quadratic equations, such as 'x squared plus 4x equals negative 3', and then solves them through factoring or other algebraic techniques.

💡Taking logarithms

Taking the logarithm of both sides is a technique used to simplify exponential equations where the base cannot be changed easily. The speaker demonstrates this when solving equations like '3 raised to x equals 8', showing how the log function helps isolate the exponent and solve for 'x'.

💡Natural logarithm

The natural logarithm, denoted ln, is the logarithm to the base 'e', where 'e' is a mathematical constant approximately equal to 2.718. In the video, the natural log is used for equations involving base 'e', like 'e raised to x equals 7'. The speaker notes that using ln simplifies equations involving the base 'e'.

💡Exponent properties

Exponent properties include rules like adding exponents when multiplying terms with the same base or subtracting exponents when dividing. The video frequently applies these properties, such as in 'two raised to the x squared times two raised to the three x equals sixteen', to simplify equations before solving them.

💡Complex fraction simplification

Complex fraction simplification refers to simplifying fractions that involve other fractions in the numerator or denominator. The speaker uses this process when solving an equation involving '3 to the negative 1', multiplying both the numerator and denominator by 3 to simplify the terms and solve the equation.

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