Use The Method Learned Today In Class By Taking The $\log$ Of Both Sides.Solve The Equation:${ 6^{\frac{x}{3}} + 20 = 43 }$

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Introduction

In mathematics, solving exponential equations can be a challenging task, especially when dealing with complex expressions. One powerful method for solving these types of equations is by using logarithms. In this article, we will explore how to use the method learned today in class by taking the log of both sides to solve the equation: 6x3+20=436^{\frac{x}{3}} + 20 = 43.

Understanding Exponential Equations

Exponential equations are equations that contain an exponential expression, which is a mathematical expression that involves a base raised to a power. In the given equation, 6x36^{\frac{x}{3}} is an exponential expression where the base is 6 and the exponent is x3\frac{x}{3}. The equation also contains a constant term, 20, which is added to the exponential expression.

The Method of Taking the Logarithm of Both Sides

One of the most effective methods for solving exponential equations is by taking the logarithm of both sides. This method involves using the logarithmic function to transform the exponential expression into a linear expression. The logarithmic function is a mathematical function that takes a positive real number as input and returns a real number as output.

To apply this method, we need to choose a base for the logarithm. In this case, we will use the common logarithm, which has a base of 10. We will also use the property of logarithms that states: log⁡b(ac)=clog⁡b(a)\log_b (a^c) = c \log_b (a).

Applying the Method to the Given Equation

Now, let's apply the method of taking the logarithm of both sides to the given equation: 6x3+20=436^{\frac{x}{3}} + 20 = 43.

First, we will subtract 20 from both sides of the equation to isolate the exponential expression:

6x3=43−206^{\frac{x}{3}} = 43 - 20

6x3=236^{\frac{x}{3}} = 23

Next, we will take the logarithm of both sides of the equation using the common logarithm:

log⁡10(6x3)=log⁡10(23)\log_{10} (6^{\frac{x}{3}}) = \log_{10} (23)

Using the property of logarithms, we can rewrite the left-hand side of the equation as:

x3log⁡10(6)=log⁡10(23)\frac{x}{3} \log_{10} (6) = \log_{10} (23)

Solving for x

Now, we can solve for x by multiplying both sides of the equation by 3:

xlog⁡10(6)=3log⁡10(23)x \log_{10} (6) = 3 \log_{10} (23)

x=3log⁡10(23)log⁡10(6)x = \frac{3 \log_{10} (23)}{\log_{10} (6)}

Evaluating the Expression

To evaluate the expression, we need to calculate the values of the logarithms. Using a calculator, we get:

log⁡10(23)≈1.3611\log_{10} (23) \approx 1.3611

log⁡10(6)≈0.7782\log_{10} (6) \approx 0.7782

Now, we can substitute these values into the expression:

x≈3×1.36110.7782x \approx \frac{3 \times 1.3611}{0.7782}

x≈4.08330.7782x \approx \frac{4.0833}{0.7782}

x≈5.25x \approx 5.25

Conclusion

In this article, we learned how to use the method of taking the logarithm of both sides to solve the equation: 6x3+20=436^{\frac{x}{3}} + 20 = 43. We applied this method by first isolating the exponential expression, then taking the logarithm of both sides, and finally solving for x. The solution to the equation is x ≈ 5.25.

Final Answer

The final answer is: 5.25\boxed{5.25}

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Q: What is the main concept behind solving exponential equations with logarithms?

A: The main concept behind solving exponential equations with logarithms is to use the property of logarithms that states: log⁡b(ac)=clog⁡b(a)\log_b (a^c) = c \log_b (a). This property allows us to transform the exponential expression into a linear expression, making it easier to solve for the variable.

Q: Why do we need to choose a base for the logarithm when solving exponential equations?

A: We need to choose a base for the logarithm because the logarithmic function is defined only for positive real numbers. By choosing a base, we can ensure that the logarithm is defined and can be used to solve the equation.

Q: Can we use any base for the logarithm when solving exponential equations?

A: No, we cannot use any base for the logarithm. The base of the logarithm must be a positive real number. In this article, we used the common logarithm, which has a base of 10.

Q: How do we apply the method of taking the logarithm of both sides to solve an exponential equation?

A: To apply the method of taking the logarithm of both sides, we need to follow these steps:

  1. Isolate the exponential expression on one side of the equation.
  2. Take the logarithm of both sides of the equation using the chosen base.
  3. Use the property of logarithms to rewrite the left-hand side of the equation as a linear expression.
  4. Solve for the variable.

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

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

  • Not isolating the exponential expression on one side of the equation.
  • Not choosing a base for the logarithm.
  • Not using the property of logarithms to rewrite the left-hand side of the equation.
  • Not solving for the variable correctly.

Q: Can we use logarithms to solve exponential equations with fractional exponents?

A: Yes, we can use logarithms to solve exponential equations with fractional exponents. The method of taking the logarithm of both sides can be applied to solve these types of equations.

Q: How do we evaluate the expression after taking the logarithm of both sides?

A: To evaluate the expression after taking the logarithm of both sides, we need to calculate the values of the logarithms using a calculator. We can then substitute these values into the expression to find the solution.

Q: What is the final answer to the equation 6x3+20=436^{\frac{x}{3}} + 20 = 43?

A: The final answer to the equation 6x3+20=436^{\frac{x}{3}} + 20 = 43 is x ≈ 5.25.

Q: Can you provide more examples of solving exponential equations with logarithms?

A: Yes, here are a few more examples of solving exponential equations with logarithms:

  • 2x+10=502^x + 10 = 50
  • 3x2−5=203^{\frac{x}{2}} - 5 = 20
  • 4x−15=254^x - 15 = 25

These examples demonstrate how to apply the method of taking the logarithm of both sides to solve exponential equations with different bases and exponents.