Solve The Equation. Round To The Nearest Ten-thousandth.${ 7^y = 15 }$ { Y = \square \}

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the equation 7y=15{7^y = 15} and finding the value of y{y}. We will use a step-by-step approach to ensure that we understand the process and can apply it to similar problems.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential expression, which is a number raised to a power. In the equation 7y=15{7^y = 15}, the base is 7, and the exponent is y{y}. The equation states that 7 raised to the power of y{y} is equal to 15.

Solving Exponential Equations

To solve exponential equations, we need to isolate the variable, which in this case is y{y}. We can do this by using logarithms. A logarithm is the inverse operation of exponentiation. In other words, if ab=c{a^b = c}, then log⁡ac=b{\log_a c = b}.

Using Logarithms to Solve Exponential Equations

We can use logarithms to solve the equation 7y=15{7^y = 15} by taking the logarithm of both sides. This will allow us to isolate y{y}.

log⁡7y=log⁡15{ \log 7^y = \log 15 }

Using the property of logarithms that states log⁡ab=blog⁡a{\log a^b = b \log a}, we can rewrite the equation as:

ylog⁡7=log⁡15{ y \log 7 = \log 15 }

Now, we can divide both sides by log⁡7{\log 7} to isolate y{y}:

y=log⁡15log⁡7{ y = \frac{\log 15}{\log 7} }

Using a Calculator to Find the Value of y{y}

To find the value of y{y}, we can use a calculator to evaluate the expression log⁡15log⁡7{\frac{\log 15}{\log 7}}. This will give us the value of y{y} to the nearest ten-thousandth.

y≈1.1760912590.84509804{ y \approx \frac{1.176091259}{0.84509804} }

y≈1.389{ y \approx 1.389 }

Rounding to the Nearest Ten-Thousandth

The problem asks us to round the value of y{y} to the nearest ten-thousandth. This means that we need to round the value of y{y} to four decimal places.

y≈1.3890{ y \approx 1.3890 }

Conclusion

Solving exponential equations requires a step-by-step approach and the use of logarithms. By following the steps outlined in this article, we can solve the equation 7y=15{7^y = 15} and find the value of y{y}. We can use a calculator to evaluate the expression and round the value of y{y} to the nearest ten-thousandth.

Frequently Asked Questions

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable by using logarithms.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if ab=c{a^b = c}, then log⁡ac=b{\log_a c = b}.

Q: How do I use a calculator to find the value of y{y}?

A: To find the value of y{y}, you can use a calculator to evaluate the expression log⁡15log⁡7{\frac{\log 15}{\log 7}}.

Further Reading

If you want to learn more about exponential equations and logarithms, here are some additional resources:

  • Khan Academy: Exponential Equations
  • Mathway: Exponential Equations
  • Wolfram Alpha: Exponential Equations

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline

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Introduction

Exponential equations can be a challenging topic for many students and professionals. In this article, we will provide answers to some of the most frequently asked questions about exponential equations.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, the equation 2x=8{2^x = 8} is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable by using logarithms. This involves taking the logarithm of both sides of the equation and then using the properties of logarithms to simplify the expression.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if ab=c{a^b = c}, then log⁡ac=b{\log_a c = b}. For example, if 23=8{2^3 = 8}, then log⁡28=3{\log_2 8 = 3}.

Q: How do I use a calculator to find the value of y{y}?

A: To find the value of y{y}, you can use a calculator to evaluate the expression log⁡15log⁡7{\frac{\log 15}{\log 7}}. This will give you the value of y{y} to the nearest ten-thousandth.

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

A: A logarithmic equation is an equation that involves a logarithmic expression, which is the inverse of an exponential expression. For example, the equation log⁡2x=3{\log_2 x = 3} is a logarithmic equation. An exponential equation, on the other hand, involves an exponential expression, such as 2x=8{2^x = 8}.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph by hand.

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers. This means that the function is defined for all values of x{x}.

Q: What is the range of an exponential function?

A: The range of an exponential function depends on the base of the function. If the base is greater than 1, the range is all positive real numbers. If the base is between 0 and 1, the range is all non-positive real numbers.

Q: How do I find the inverse of an exponential function?

A: To find the inverse of an exponential function, you can swap the roles of x{x} and y{y} and then solve for y{y}. This will give you the inverse function.

Q: What is the difference between an exponential function and a power function?

A: An exponential function is a function of the form f(x)=ax{f(x) = a^x}, where a{a} is a positive constant. A power function, on the other hand, is a function of the form f(x)=xn{f(x) = x^n}, where n{n} is a constant.

Conclusion

Exponential equations can be a challenging topic, but with practice and patience, you can become proficient in solving them. Remember to use logarithms to isolate the variable and to use a calculator to evaluate expressions. If you have any further questions, feel free to ask.

Frequently Asked Questions (FAQs)

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

A: A logarithmic equation is an equation that involves a logarithmic expression, which is the inverse of an exponential expression. An exponential equation, on the other hand, involves an exponential expression.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph by hand.

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers.

Q: What is the range of an exponential function?

A: The range of an exponential function depends on the base of the function.

Further Reading

If you want to learn more about exponential equations and logarithms, here are some additional resources:

  • Khan Academy: Exponential Equations
  • Mathway: Exponential Equations
  • Wolfram Alpha: Exponential Equations

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline