Select The Correct Answer.What Is The Value Of $x$ In $2\left(5^x\right)=14$?A. $ Log ⁡ 5 Log ⁡ 7 \frac{\log 5}{\log 7} L O G 7 L O G 5 ​ [/tex] B. $\log 7-\log 5$ C. $\frac{\log 7}{\log 5}$ D. $ Log ⁡ 2 \log 2 Lo G 2 [/tex]

by ADMIN 236 views

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithms and their properties. In this article, we will explore how to solve exponential equations, with a focus on the given problem: $2\left(5^x\right)=14$. We will break down the solution into manageable steps, using logarithmic properties to isolate the variable $x$.

Understanding Exponential Equations

An exponential equation is an equation in which the variable is raised to a power. In the given problem, $2\left(5^x\right)=14$, the variable $x$ is raised to the power of 5. Exponential equations can be solved using logarithmic properties, which allow us to isolate the variable and find its value.

Step 1: Isolate the Exponential Term

The first step in solving the exponential equation is to isolate the exponential term. In this case, we can start by dividing both sides of the equation by 2:

2(5x)2=142\frac{2\left(5^x\right)}{2}=\frac{14}{2}

This simplifies to:

5x=75^x=7

Step 2: Use Logarithmic Properties

Now that we have isolated the exponential term, we can use logarithmic properties to solve for $x$. We will use the property that $\log_a{b}=c$ is equivalent to $a^c=b$. In this case, we can take the logarithm of both sides of the equation:

log⁡55x=log⁡57\log_5{5^x}=\log_5{7}

Using the property that $\log_a{a^c}=c$, we can simplify the left-hand side of the equation:

x=log⁡57x=\log_5{7}

Step 3: Simplify the Expression

The expression $\log_5{7}$ is a logarithmic expression, and we can simplify it using the change-of-base formula:

log⁡57=log⁡7log⁡5\log_5{7}=\frac{\log{7}}{\log{5}}

This expression is equivalent to the answer choice C.

Conclusion

In this article, we have solved the exponential equation $2\left(5^x\right)=14$ using logarithmic properties. We have broken down the solution into manageable steps, isolating the exponential term and using logarithmic properties to solve for $x$. The final answer is:

log⁡7log⁡5\frac{\log 7}{\log 5}

This answer is equivalent to answer choice C.

Answer Key

  • A. $\frac{\log 5}{\log 7}$
  • B. $\log 7-\log 5$
  • C. $\frac{\log 7}{\log 5}$
  • D. $\log 2$

The correct answer is C. $\frac{\log 7}{\log 5}$.

Discussion

This problem requires a deep understanding of logarithmic properties and their applications in solving exponential equations. The solution involves breaking down the problem into manageable steps, isolating the exponential term, and using logarithmic properties to solve for $x$. The final answer is a logarithmic expression, which can be simplified using the change-of-base formula.

Related Topics

  • Exponential equations
  • Logarithmic properties
  • Change-of-base formula
  • Solving equations using logarithms

Practice Problems

  • Solve the exponential equation $3\left(2^x\right)=27$.
  • Solve the exponential equation $4\left(3^x\right)=64$.
  • Solve the exponential equation $2\left(4^x\right)=32$.

Q: What is an exponential equation?

A: An exponential equation is an equation in which the variable is raised to a power. For example, $2^x=8$ is an exponential equation, where the variable $x$ is raised to the power of 2.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use logarithmic properties to isolate the variable. For example, to solve the equation $2^x=8$, you can take the logarithm of both sides:

log⁡22x=log⁡28\log_2{2^x}=\log_2{8}

Using the property that $\log_a{a^c}=c$, you can simplify the left-hand side of the equation:

x=log⁡28x=\log_2{8}

Q: What is the change-of-base formula?

A: The change-of-base formula is a formula that allows you to change the base of a logarithm. For example, if you have a logarithm in base 2, you can change it to base 10 using the formula:

log⁡2x=log⁡10xlog⁡102\log_2{x}=\frac{\log_{10}{x}}{\log_{10}{2}}

Q: How do I use the change-of-base formula?

A: To use the change-of-base formula, you can substitute the expression you want to simplify into the formula. For example, to simplify the expression $\log_2{8}$, you can use the change-of-base formula:

log⁡28=log⁡108log⁡102\log_2{8}=\frac{\log_{10}{8}}{\log_{10}{2}}

Using a calculator, you can evaluate the expression to get:

log⁡28=3\log_2{8}=3

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation in which the variable is the exponent of a power. For example, $\log_2{x}=3$ is a logarithmic equation, where the variable $x$ is the base of the power. An exponential equation, on the other hand, is an equation in which the variable is raised to a power. For example, $2^x=8$ is an exponential equation, where the variable $x$ is raised to the power of 2.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the property that $\log_a{a^c}=c$. For example, to solve the equation $\log_2{x}=3$, you can use the property to simplify the equation:

x=23x=2^3

Using a calculator, you can evaluate the expression to get:

x=8x=8

Q: What are some common mistakes to avoid when solving exponential and logarithmic equations?

A: Some common mistakes to avoid when solving exponential and logarithmic equations include:

  • Not using the correct base for the logarithm
  • Not using the change-of-base formula when necessary
  • Not simplifying the expression correctly
  • Not checking the domain of the logarithm

Q: How do I check the domain of a logarithm?

A: To check the domain of a logarithm, you can check that the base of the logarithm is positive and not equal to 1. For example, the domain of the logarithm $\log_2{x}$ is all real numbers greater than 0, since the base 2 is positive and not equal to 1.

Q: What are some real-world applications of exponential and logarithmic equations?

A: Exponential and logarithmic equations have many real-world applications, including:

  • Modeling population growth and decay
  • Modeling chemical reactions and rates of reaction
  • Modeling financial growth and decay
  • Modeling electrical circuits and signal processing

These are just a few examples of the many real-world applications of exponential and logarithmic equations.