Solve $\log_2 X = \log_5 3 + 1$ By Graphing. Round To The Nearest Tenth.$x = \square$

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Introduction

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Logarithmic equations can be challenging to solve, especially when they involve different bases. In this article, we will explore how to solve a logarithmic equation using graphing. We will use the equation $\log_2 x = \log_5 3 + 1$ as an example and round the solution to the nearest tenth.

Understanding Logarithmic Equations

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A logarithmic equation is an equation that involves a logarithm. The general form of a logarithmic equation is $\log_b a = c$, where $b$ is the base, $a$ is the argument, and $c$ is the result. In this equation, the logarithm is the inverse of the exponential function.

The Equation $\log_2 x = \log_5 3 + 1$

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The given equation is $\log_2 x = \log_5 3 + 1$. To solve this equation, we need to first simplify the right-hand side. We can use the change of base formula to rewrite $\log_5 3$ in terms of base 2.

Change of Base Formula

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The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number. We can use this formula to rewrite $\log_5 3$ in terms of base 2.

$\log_5 3 = \frac{\log_2 3}{\log_2 5}$

Simplifying the Equation

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Now that we have rewritten $\log_5 3$ in terms of base 2, we can simplify the equation.

$\log_2 x = \frac{\log_2 3}{\log_2 5} + 1$

Graphing the Equation

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To solve the equation using graphing, we need to graph the two functions $y = \log_2 x$ and $y = \frac{\log_2 3}{\log_2 5} + 1$. We can use a graphing calculator or software to graph these functions.

Graphing $y = \log_2 x$

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The graph of $y = \log_2 x$ is a logarithmic curve that opens upwards. The x-axis is the asymptote of the graph.

Graphing $y = \frac{\log_2 3}{\log_2 5} + 1$

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The graph of $y = \frac{\log_2 3}{\log_2 5} + 1$ is a horizontal line that is shifted up by 1 unit.

Finding the Intersection Point

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To find the solution to the equation, we need to find the intersection point of the two graphs. The intersection point is the point where the two graphs meet.

Using a Graphing Calculator

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We can use a graphing calculator to find the intersection point. To do this, we need to enter the two functions into the calculator and use the "intersect" function to find the intersection point.

Finding the Solution

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The intersection point is approximately (x, y) = (5.4, 2.1). We can round the x-coordinate to the nearest tenth to get the solution.

Conclusion

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In this article, we have shown how to solve a logarithmic equation using graphing. We used the equation $\log_2 x = \log_5 3 + 1$ as an example and rounded the solution to the nearest tenth. The solution is x = 5.4.

Final Answer

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The final answer is $\boxed{5.4}$.

References

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• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Graphing Logarithmic Functions" by Math Is Fun
• [3] "Change of Base Formula" by Wolfram MathWorld
  

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Introduction

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In our previous article, we explored how to solve a logarithmic equation using graphing. We used the equation $\log_2 x = \log_5 3 + 1$ as an example and rounded the solution to the nearest tenth. In this article, we will answer some frequently asked questions about solving logarithmic equations using graphing.

Q: What is the change of base formula?

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A: The change of base formula is a mathematical formula that allows us to rewrite a logarithm in terms of a different base. The formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number.

Q: How do I use the change of base formula to rewrite $\log_5 3$ in terms of base 2?

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A: To rewrite $\log_5 3$ in terms of base 2, we can use the change of base formula. We get $\log_5 3 = \frac{\log_2 3}{\log_2 5}$.

Q: What is the graph of $y = \log_2 x$?

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A: The graph of $y = \log_2 x$ is a logarithmic curve that opens upwards. The x-axis is the asymptote of the graph.

Q: What is the graph of $y = \frac{\log_2 3}{\log_2 5} + 1$?

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A: The graph of $y = \frac{\log_2 3}{\log_2 5} + 1$ is a horizontal line that is shifted up by 1 unit.

Q: How do I find the intersection point of the two graphs?

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A: To find the intersection point of the two graphs, we need to use a graphing calculator or software. We can enter the two functions into the calculator and use the "intersect" function to find the intersection point.

Q: What is the solution to the equation $\log_2 x = \log_5 3 + 1$?

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A: The solution to the equation $\log_2 x = \log_5 3 + 1$ is x = 5.4.

Q: Can I use graphing to solve any logarithmic equation?

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A: Yes, you can use graphing to solve any logarithmic equation. However, you need to make sure that the equation is in the form $\log_b x = c$, where $b$ is the base and $c$ is the result.

Q: What are some common mistakes to avoid when using graphing to solve logarithmic equations?

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A: Some common mistakes to avoid when using graphing to solve logarithmic equations include:

• Not using the change of base formula to rewrite the logarithm in terms of a different base.
• Not graphing the two functions correctly.
• Not finding the intersection point correctly.
• Not rounding the solution to the nearest tenth.

Q: How can I practice solving logarithmic equations using graphing?

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A: You can practice solving logarithmic equations using graphing by using online graphing calculators or software. You can also try solving different types of logarithmic equations to get more practice.

Conclusion

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In this article, we have answered some frequently asked questions about solving logarithmic equations using graphing. We hope that this article has been helpful in clarifying any doubts you may have had about solving logarithmic equations using graphing.

Final Answer

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The final answer is $\boxed{5.4}$.

References

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• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Graphing Logarithmic Functions" by Math Is Fun
• [3] "Change of Base Formula" by Wolfram MathWorld