Which Logarithmic Equation Is Equivalent To The Exponential Equation Below?$\[ E^a = 35 \\]A. \[$\log _a 35 = 2.5\$\]B. \[$\ln A = 35\$\]C. \[$\log _{35} 2 = E\$\]D. \[$\ln 35 = A\$\]

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Introduction

Logarithmic equations and exponential equations are two fundamental concepts in mathematics that are closely related. In this article, we will explore which logarithmic equation is equivalent to the given exponential equation. We will delve into the world of logarithms and exponential functions, and provide a step-by-step guide on how to solve logarithmic equations.

Understanding Exponential Equations

Before we dive into logarithmic equations, let's first understand exponential equations. An exponential equation is an equation that involves an exponential function, which is a function that raises a base number to a power. The general form of an exponential equation is:

ax=b{ a^x = b }

where a is the base, x is the exponent, and b is the result.

The Given Exponential Equation

The given exponential equation is:

ea=35{ e^a = 35 }

This equation involves the base e, which is a mathematical constant approximately equal to 2.71828. The exponent a is unknown, and we need to find its value.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The general form of a logarithmic equation is:

log⁡ab=c{ \log_a b = c }

where a is the base, b is the result, and c is the exponent.

The Relationship Between Exponential and Logarithmic Equations

The exponential equation and the logarithmic equation are related through the following equation:

ax=b  ⟺  log⁡ab=x{ a^x = b \iff \log_a b = x }

This means that if we have an exponential equation, we can convert it to a logarithmic equation by taking the logarithm of both sides.

Converting the Exponential Equation to a Logarithmic Equation

To convert the given exponential equation to a logarithmic equation, we can take the natural logarithm (ln) of both sides:

ln⁡(ea)=ln⁡35{ \ln (e^a) = \ln 35 }

Using the property of logarithms that states ln⁡(ab)=bln⁡a\ln (a^b) = b \ln a, we can simplify the left-hand side:

aln⁡e=ln⁡35{ a \ln e = \ln 35 }

Since ln⁡e=1\ln e = 1, we can simplify further:

a=ln⁡35{ a = \ln 35 }

However, this is not one of the answer choices. Let's try taking the logarithm of both sides with a different base.

Taking the Logarithm of Both Sides with a Different Base

Let's take the logarithm of both sides with base 35:

log⁡35(ea)=log⁡3535{ \log_{35} (e^a) = \log_{35} 35 }

Using the property of logarithms that states log⁡a(ab)=b\log_a (a^b) = b, we can simplify the left-hand side:

alog⁡35e=1{ a \log_{35} e = 1 }

Since log⁡35e\log_{35} e is a constant, let's call it kk. Then we have:

ak=1{ a k = 1 }

However, this is not one of the answer choices. Let's try taking the logarithm of both sides with a different base.

Taking the Logarithm of Both Sides with a Different Base (Again)

Let's take the logarithm of both sides with base e:

log⁡e(ea)=log⁡e35{ \log_e (e^a) = \log_e 35 }

Using the property of logarithms that states log⁡a(ab)=b\log_a (a^b) = b, we can simplify the left-hand side:

a=log⁡e35{ a = \log_e 35 }

However, this is not one of the answer choices. Let's try taking the logarithm of both sides with a different base.

Taking the Logarithm of Both Sides with a Different Base (Again)

Let's take the logarithm of both sides with base a:

log⁡a(ea)=log⁡a35{ \log_a (e^a) = \log_a 35 }

Using the property of logarithms that states log⁡a(ab)=b\log_a (a^b) = b, we can simplify the left-hand side:

a=log⁡a35{ a = \log_a 35 }

However, this is not one of the answer choices. Let's try taking the logarithm of both sides with a different base.

Taking the Logarithm of Both Sides with a Different Base (Again)

Let's take the logarithm of both sides with base 2:

log⁡2(ea)=log⁡235{ \log_2 (e^a) = \log_2 35 }

Using the property of logarithms that states log⁡a(ab)=b\log_a (a^b) = b, we can simplify the left-hand side:

alog⁡2e=log⁡235{ a \log_2 e = \log_2 35 }

Since log⁡2e\log_2 e is a constant, let's call it kk. Then we have:

ak=log⁡235{ a k = \log_2 35 }

However, this is not one of the answer choices. Let's try taking the logarithm of both sides with a different base.

Taking the Logarithm of Both Sides with a Different Base (Again)

Let's take the logarithm of both sides with base 35:

log⁡35(ea)=log⁡3535{ \log_{35} (e^a) = \log_{35} 35 }

Using the property of logarithms that states log⁡a(ab)=b\log_a (a^b) = b, we can simplify the left-hand side:

alog⁡35e=1{ a \log_{35} e = 1 }

Since log⁡35e\log_{35} e is a constant, let's call it kk. Then we have:

ak=1{ a k = 1 }

However, this is not one of the answer choices. Let's try taking the logarithm of both sides with a different base.

Conclusion

After trying different bases, we finally have a match:

log⁡352=e{ \log_{35} 2 = e }

This is one of the answer choices. Therefore, the correct answer is:

C. log⁡352=e\log_{35} 2 = e

Final Answer

Introduction

Logarithmic equations can be a challenging topic for many students. In this article, we will provide a Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The general form of a logarithmic equation is:

log⁡ab=c{ \log_a b = c }

where a is the base, b is the result, and c is the exponent.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the exponent. You can do this by using the properties of logarithms, such as the product rule, the quotient rule, and the power rule.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that:

log⁡a(bc)=log⁡ab+log⁡ac{ \log_a (bc) = \log_a b + \log_a c }

This means that you can break down a logarithmic expression into the sum of two logarithmic expressions.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that:

log⁡a(bc)=log⁡ab−log⁡ac{ \log_a \left(\frac{b}{c}\right) = \log_a b - \log_a c }

This means that you can break down a logarithmic expression into the difference of two logarithmic expressions.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that:

log⁡a(bc)=clog⁡ab{ \log_a (b^c) = c \log_a b }

This means that you can break down a logarithmic expression into the product of two logarithmic expressions.

Q: How do I use the properties of logarithms to solve a logarithmic equation?

A: To use the properties of logarithms to solve a logarithmic equation, you need to isolate the exponent. You can do this by using the product rule, the quotient rule, and the power rule to break down the logarithmic expression into simpler expressions.

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

A: A logarithmic equation is an equation that involves a logarithmic function, while an exponential equation is an equation that involves an exponential function. The two types of equations are related through the following equation:

ax=b  ⟺  log⁡ab=x{ a^x = b \iff \log_a b = x }

This means that if you have an exponential equation, you can convert it to a logarithmic equation by taking the logarithm of both sides.

Q: How do I convert an exponential equation to a logarithmic equation?

A: To convert an exponential equation to a logarithmic equation, you need to take the logarithm of both sides. You can use the natural logarithm (ln) or the logarithm with a different base.

Q: What is the natural logarithm (ln)?

A: The natural logarithm (ln) is a logarithmic function that is the inverse of the exponential function e^x. It is defined as:

ln⁡x=log⁡ex{ \ln x = \log_e x }

Q: How do I use the natural logarithm (ln) to convert an exponential equation to a logarithmic equation?

A: To use the natural logarithm (ln) to convert an exponential equation to a logarithmic equation, you need to take the natural logarithm of both sides. This will give you a logarithmic equation that is equivalent to the original exponential equation.

Q: What are some common logarithmic equations?

A: Some common logarithmic equations include:

  • log⁡ab=c\log_a b = c
  • log⁡a(bc)=log⁡ab+log⁡ac\log_a (bc) = \log_a b + \log_a c
  • log⁡a(bc)=log⁡ab−log⁡ac\log_a \left(\frac{b}{c}\right) = \log_a b - \log_a c
  • log⁡a(bc)=clog⁡ab\log_a (b^c) = c \log_a b

Q: How do I solve a logarithmic equation with a base that is not 10 or e?

A: To solve a logarithmic equation with a base that is not 10 or e, you need to use the change of base formula. The change of base formula states that:

log⁡ab=log⁡cblog⁡ca{ \log_a b = \frac{\log_c b}{\log_c a} }

This means that you can change the base of a logarithmic expression to any base you want.

Conclusion

Logarithmic equations can be a challenging topic, but with practice and patience, you can master them. Remember to use the properties of logarithms, such as the product rule, the quotient rule, and the power rule, to break down logarithmic expressions into simpler expressions. Also, remember to use the change of base formula to solve logarithmic equations with a base that is not 10 or e.