How to Solve Exponential Equations?

[Alexeia]

Hi,

I attached a pdf with a question and its answers. I don't understand the whole thing basically, e.g what their asking for..

1.2.1;1.2.2;1.2.3

1.2.2 - How many solutions for x will the equation have..?
1.2.3 - Largest integer for which it will have solutions..?

Any explanations will help..

Thanks

 

[MarkFL]

Here are the questions:

1.2 Given: \(\displaystyle 2^x+2^{x+2}=-5y+20\)

1.2.1 Express \(\displaystyle 2^x\) in terms of $y$.

Here, you are expected to solve the equation for $2^x$. Recall the property of exponents:

\(\displaystyle a^{b+c}=a^ba^c\)

which means:

\(\displaystyle 2^{x+2}=2^x2^2=4\cdot2^x\)

So now the equation becomes:

\(\displaystyle 2^x+4\cdot2^x=-5y+20\)

Can you continue? Combine the two like terms on the left side as your first step.

After we get this part solved, where you understand it, we can move on to the remaining parts, as they depend on this part.

 

[Alexeia]

Here are the questions:

1.2 Given: \(\displaystyle 2^x+2^{x+2}=-5y+20\)

1.2.1 Express \(\displaystyle 2^x\) in terms of $y$.

Here, you are expected to solve the equation for $2^x$. Recall the property of exponents:

\(\displaystyle a^{b+c}=a^ba^c\)

which means:

\(\displaystyle 2^{x+2}=2^x2^2=4\cdot2^x\)

So now the equation becomes:

\(\displaystyle 2^x+4\cdot2^x=-5y+20\)

Can you continue? Combine the two like terms on the left side as your first step.

After we get this part solved, where you understand it, we can move on to the remaining parts, as they depend on this part.

Awesome, the way the questioned was asked confused me.. Another way of saying it then is - Solve for 2^x.. But it doesn't yet mean to solve for x right?

Then Question 1.2.3, can you shed some light on that one pls?

 

[Evgeny.Makarov]

Then Question 1.2.3, can you shed some light on that one pls?

1.2 Given:
\[
2^x+2^{x+2}=-5y+20\tag{1}
\]
...
1.2.3: Solve for $x$ if $y$ is the largest possible integer value for which (1) will have solutions.

Question 1.2.3 (admittedly, a bit convoluted) says the following. Let $P(y)$ be some property of $y$ (to be discussed later). Find the largest integer $y$ such that $P(y)$ holds and call it $y_{\text{max}}$. Now solve (1) for $x$ assuming that $y=y_{\text{max}}$.

The property $P(y)$ in question is that (1) has a solution $x$ for a given $y$. Note that the equation
\[
2^x+2^{x+2}=z
\]
has some solution $x$ iff $z>0$. Therefore,
\[
P(y)\text{ holds }\iff -5y+20>0 \iff y<4.
\]
The largest $y_{\text{max}}$ satisfying this property is 3 (since the inequality is strict). Therefore, you are supposed to solve (1) when $y=3$.

 

[CellCoree]

for reaching out. Exponential equations are mathematical equations that involve an unknown variable in the exponent, such as 2^x or e^x. They are commonly used in science and mathematics to model growth and decay processes.

In the attached pdf, the questions are asking about the solutions to a specific exponential equation. In 1.2.2, the question is asking how many different values of x will satisfy the equation. This means that the equation will have multiple solutions, or values of x, that make the equation true. In 1.2.3, the question is asking for the largest integer value of x that will satisfy the equation. This means that the solutions to the equation will be limited to whole numbers, or integers.

To solve these types of equations, you can use algebraic methods such as substitution or logarithms. It's important to carefully read the question and identify what is being asked for in order to find the correct solutions.

I hope this helps to clarify the concepts of exponential equations. Please let me know if you have any further questions. Keep up the good work in your studies!