If Log ⁡ 2 ( 3 X + 5 ) = 3 \log _2(3x+5)=3 Lo G 2 ​ ( 3 X + 5 ) = 3 , Then X = X= X = □ \square □ You May Enter The Exact Value Or Round To 4 Decimal Places.

Introduction

In this article, we will explore the concept of logarithms and how to solve equations involving logarithms. We will use the given equation $\log _2(3x+5)=3$ to find the value of $x$. This equation involves a logarithm with base 2, and we will use properties of logarithms to solve for $x$.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if $a^b=c$, then $\log _a(c)=b$. The logarithm with base $a$ of a number $c$ is the exponent to which $a$ must be raised to produce $c$. For example, $\log _2(8)=3$ because $2^3=8$.

Solving the Equation

To solve the equation $\log _2(3x+5)=3$, we can start by rewriting the equation in exponential form. This means that we can rewrite the equation as $2^3=3x+5$. We can simplify this equation by evaluating the left-hand side: $8=3x+5$.

Isolating $x$

Next, we can isolate $x$ by subtracting 5 from both sides of the equation: $8-5=3x$. This simplifies to $3=3x$. We can then divide both sides of the equation by 3 to solve for $x$: $x=1$.

Verifying the Solution

To verify that $x=1$ is a solution to the equation, we can substitute $x=1$ into the original equation: $\log _2(3(1)+5)=\log _2(8)=3$. This shows that $x=1$ is indeed a solution to the equation.

Conclusion

In this article, we used the properties of logarithms to solve the equation $\log _2(3x+5)=3$. We started by rewriting the equation in exponential form and then isolated $x$ by subtracting 5 from both sides of the equation. We then divided both sides of the equation by 3 to solve for $x$. We verified that $x=1$ is a solution to the equation by substituting $x=1$ into the original equation.

Additional Examples

Here are a few additional examples of solving equations involving logarithms:

• $\log _2(2x+1)=2$: This equation can be rewritten as $2^2=2x+1$, which simplifies to $4=2x+1$. We can then isolate $x$ by subtracting 1 from both sides of the equation: $4-1=2x$. This simplifies to $3=2x$. We can then divide both sides of the equation by 2 to solve for $x$: $x=\frac{3}{2}$.
• $\log _3(x+2)=1$: This equation can be rewritten as $3^1=x+2$, which simplifies to $3=x+2$. We can then isolate $x$ by subtracting 2 from both sides of the equation: $3-2=x$. This simplifies to $x=1$.
• $\log _4(2x-1)=2$: This equation can be rewritten as $4^2=2x-1$, which simplifies to $16=2x-1$. We can then isolate $x$ by adding 1 to both sides of the equation: $16+1=2x$. This simplifies to $17=2x$. We can then divide both sides of the equation by 2 to solve for $x$: $x=\frac{17}{2}$.

Tips and Tricks

Here are a few tips and tricks for solving equations involving logarithms:

• Always start by rewriting the equation in exponential form.
• Use the properties of logarithms to simplify the equation.
• Isolate $x$ by subtracting or adding terms from both sides of the equation.
• Divide both sides of the equation by the coefficient of $x$ to solve for $x$.
• Verify that the solution is correct by substituting it into the original equation.

Conclusion

In this article, we used the properties of logarithms to solve the equation $\log _2(3x+5)=3$. We started by rewriting the equation in exponential form and then isolated $x$ by subtracting 5 from both sides of the equation. We then divided both sides of the equation by 3 to solve for $x$. We verified that $x=1$ is a solution to the equation by substituting $x=1$ into the original equation. We also provided additional examples of solving equations involving logarithms and offered tips and tricks for solving these types of equations.

Introduction

In our previous article, we explored the concept of logarithms and how to solve equations involving logarithms. We used the given equation $\log _2(3x+5)=3$ to find the value of $x$. In this article, we will answer some common questions related to solving equations involving logarithms.

Q&A

Q: What is the definition of a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if $a^b=c$, then $\log _a(c)=b$. The logarithm with base $a$ of a number $c$ is the exponent to which $a$ must be raised to produce $c$.

Q: How do I rewrite an equation in exponential form?

A: To rewrite an equation in exponential form, you need to use the definition of a logarithm. For example, if you have the equation $\log _2(3x+5)=3$, you can rewrite it in exponential form as $2^3=3x+5$.

Q: How do I isolate $x$ in an equation involving logarithms?

A: To isolate $x$ in an equation involving logarithms, you need to use the properties of logarithms to simplify the equation. For example, if you have the equation $\log _2(3x+5)=3$, you can isolate $x$ by subtracting 5 from both sides of the equation: $8=3x$. You can then divide both sides of the equation by 3 to solve for $x$: $x=1$.

Q: What are some common mistakes to avoid when solving equations involving logarithms?

A: Some common mistakes to avoid when solving equations involving logarithms include:

• Not rewriting the equation in exponential form
• Not using the properties of logarithms to simplify the equation
• Not isolating $x$ correctly
• Not verifying the solution by substituting it into the original equation

Q: How do I verify that a solution is correct?

A: To verify that a solution is correct, you need to substitute it into the original equation. For example, if you have the equation $\log _2(3x+5)=3$ and you think that $x=1$ is a solution, you can substitute $x=1$ into the original equation: $\log _2(3(1)+5)=\log _2(8)=3$. This shows that $x=1$ is indeed a solution to the equation.

Q: What are some tips and tricks for solving equations involving logarithms?

A: Some tips and tricks for solving equations involving logarithms include:

• Always start by rewriting the equation in exponential form
• Use the properties of logarithms to simplify the equation
• Isolate $x$ by subtracting or adding terms from both sides of the equation
• Divide both sides of the equation by the coefficient of $x$ to solve for $x$
• Verify that the solution is correct by substituting it into the original equation

Conclusion

In this article, we answered some common questions related to solving equations involving logarithms. We provided definitions, examples, and tips and tricks for solving these types of equations. We also emphasized the importance of verifying that a solution is correct by substituting it into the original equation.

Additional Resources

If you are interested in learning more about logarithms and how to solve equations involving logarithms, here are some additional resources:

• Khan Academy: Logarithms
• Mathway: Logarithms
• Wolfram Alpha: Logarithms

Conclusion

In conclusion, solving equations involving logarithms can be a challenging but rewarding topic. By understanding the properties of logarithms and using the tips and tricks provided in this article, you can become proficient in solving these types of equations. Remember to always verify that a solution is correct by substituting it into the original equation.