What Is The Value Of $x$ If $e^{3x+6}=8$?A. $x=\frac{\ln 8-6}{3}$ B. $ X = Ln 6 − 8 3 X=\frac{\ln 6-8}{3} X = 3 L N 6 − 8 [/tex] C. $x=\frac{\ln \theta+6}{3}$ D. $x=\frac{\ln 6+8}{3}$
Introduction
In this article, we will explore the concept of exponential functions and how to solve equations involving them. We will use the given equation $e^{3x+6}=8$ to find the value of $x$. This equation involves the natural exponential function, which is a fundamental concept in mathematics.
Understanding Exponential Functions
Exponential functions are a type of function that can be written in the form $f(x) = a^x$, where $a$ is a positive constant. The natural exponential function, denoted by $e^x$, is a special type of exponential function where $a = e$, the base of the natural logarithm. The natural exponential function has several important properties, including:
- The exponential function is always positive: $e^x > 0$ for all real numbers $x$.
- The exponential function is continuous: $e^x$ is a continuous function, meaning that it can be drawn without lifting the pencil from the paper.
- The exponential function is increasing: $e^x$ is an increasing function, meaning that as $x$ increases, $e^x$ also increases.
Solving the Equation
To solve the equation $e^{3x+6}=8$, we can use the following steps:
- Take the natural logarithm of both sides: This will allow us to use the properties of logarithms to simplify the equation.
- Use the property of logarithms that states $\ln a^b = b \ln a$: This will allow us to simplify the equation further.
- Solve for $x$: Once we have simplified the equation, we can solve for $x$.
Step 1: Take the Natural Logarithm of Both Sides
Taking the natural logarithm of both sides of the equation gives us:
Step 2: Use the Property of Logarithms
Using the property of logarithms that states $\ln a^b = b \ln a$, we can simplify the equation further:
Step 3: Simplify the Equation
Since $\ln e = 1$, we can simplify the equation further:
Step 4: Solve for $x$
Subtracting 6 from both sides of the equation gives us:
Dividing both sides of the equation by 3 gives us:
Conclusion
In this article, we have solved the equation $e^{3x+6}=8$ to find the value of $x$. We used the properties of logarithms to simplify the equation and then solved for $x$. The final answer is:
This is option A in the given multiple-choice question.
Discussion
The equation $e^{3x+6}=8$ is a classic example of an exponential equation. The natural exponential function is a fundamental concept in mathematics, and understanding its properties is essential for solving equations involving exponential functions. In this article, we have used the properties of logarithms to simplify the equation and then solved for $x$. This is a common technique used to solve exponential equations.
Final Answer
The final answer is:
x = \frac{\ln 8 - 6}{3}$<br/>
# Q&A: Solving Exponential Equations
In our previous article, we solved the equation $e^{3x+6}=8$ to find the value of $x$. In this article, we will answer some common questions related to solving exponential equations. A: An exponential function is a function that can be written in the form $f(x) = a^x$, where $a$ is a positive constant. A logarithmic function is the inverse of an exponential function, and it can be written in the form $f(x) = \log_a x$, where $a$ is a positive constant. A: To solve an exponential equation, you can use the following steps: A: This property of logarithms states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. In other words, $\ln a^b = b \ln a$. A: To use the property of logarithms to simplify an exponential equation, you can follow these steps: A: The final answer to the equation $e^{3x+6}=8$ is: x = \frac{\ln 8 - 6}{3}
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<h2>Q: What is the difference between the natural logarithm and the common logarithm?</h2>
<p>A: The natural logarithm is the logarithm to the base $e$, and it is denoted by $\ln x$. The common logarithm is the logarithm to the base $10$, and it is denoted by $\log x$.</p>
<h2>Q: How do I convert between the natural logarithm and the common logarithm?</h2>
<p>A: To convert between the natural logarithm and the common logarithm, you can use the following formula:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ln</mi><mo></mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>log</mi><mo></mo><mi>x</mi></mrow><mrow><mi>log</mi><mo></mo><mi>e</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\ln x = \frac{\log x}{\log e}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h2>Conclusion</h2>
<p>In this article, we have answered some common questions related to solving exponential equations. We have discussed the properties of logarithms and how to use them to simplify exponential equations. We have also discussed the difference between the natural logarithm and the common logarithm, and how to convert between them.</p>
<h2>Final Answer</h2>
<p>The final answer to the equation $e^{3x+6}=8$ is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo></mo><mn>8</mn><mo>−</mo><mn>6</mn></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{\ln 8 - 6}{3}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
Introduction
Q: What is the difference between an exponential function and a logarithmic function?
Q: How do I solve an exponential equation?
Q: What is the property of logarithms that states $\ln a^b = b \ln a$?
Q: How do I use the property of logarithms to simplify an exponential equation?
Q: What is the final answer to the equation $e^{3x+6}=8$?