A) If $5^x=\frac{1}{125}$, Find The Value Of $x$.b) Find The Value Of $3^{-3} \times 10^{-3}$.

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Introduction

Exponential equations and negative exponents are fundamental concepts in mathematics that play a crucial role in various fields, including science, engineering, and economics. In this article, we will explore two problems related to these concepts: solving an exponential equation and evaluating a product of negative exponents.

Problem a: Solving Exponential Equations

If $5^x=\frac{1}{125}$, find the value of $x$.

To solve this problem, we need to use the properties of exponents and logarithms. We can start by rewriting the equation in a more manageable form.

5x=11255^x=\frac{1}{125}

We can rewrite $\frac{1}{125}$ as $5^{-3}$, since $125=5^3$.

5x=5βˆ’35^x=5^{-3}

Now, we can equate the exponents, since the bases are the same.

x=βˆ’3x=-3

Therefore, the value of $x$ is $-3$.

Problem b: Evaluating Negative Exponents

Find the value of $3^{-3} \times 10^{-3}$.

To evaluate this expression, we need to use the properties of negative exponents. We can start by rewriting each term separately.

3βˆ’3=133=1273^{-3}=\frac{1}{3^3}=\frac{1}{27}

10βˆ’3=1103=1100010^{-3}=\frac{1}{10^3}=\frac{1}{1000}

Now, we can multiply the two fractions together.

127Γ—11000=127000\frac{1}{27} \times \frac{1}{1000}=\frac{1}{27000}

Therefore, the value of $3^{-3} \times 10^{-3}$ is $\frac{1}{27000}$.

Properties of Exponents

Exponents are a fundamental concept in mathematics that play a crucial role in various fields. In this section, we will explore some of the key properties of exponents.

Product of Powers

When we multiply two powers with the same base, we can add the exponents.

amΓ—an=am+na^m \times a^n = a^{m+n}

For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Power of a Power

When we raise a power to another power, we can multiply the exponents.

(am)n=amΓ—n(a^m)^n = a^{m \times n}

For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.

Negative Exponents

A negative exponent is defined as the reciprocal of the positive exponent.

aβˆ’m=1ama^{-m} = \frac{1}{a^m}

For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Logarithms

Logarithms are the inverse of exponents. They are used to solve equations involving exponents.

Definition of Logarithm

The logarithm of a number $x$ to the base $a$ is defined as the exponent to which $a$ must be raised to produce $x$.

log⁑ax=yβ€…β€ŠβŸΊβ€…β€Šay=x\log_a x = y \iff a^y = x

For example, $\log_2 8 = 3$, since $2^3 = 8$.

Properties of Logarithms

Logarithms have several important properties that are used to solve equations involving exponents.

Product of Logarithms

When we multiply two logarithms with the same base, we can add the exponents.

log⁑a(xΓ—y)=log⁑ax+log⁑ay\log_a (x \times y) = \log_a x + \log_a y

For example, $\log_2 (4 \times 8) = \log_2 4 + \log_2 8 = 2 + 3 = 5$.

Power of Logarithm

When we raise a logarithm to another power, we can multiply the exponents.

(log⁑ax)n=log⁑a(xn)(\log_a x)^n = \log_a (x^n)

For example, $(\log_2 8)^3 = \log_2 (8^3) = \log_2 512 = 9$.

Conclusion

In this article, we have explored two problems related to exponential equations and negative exponents. We have also discussed some of the key properties of exponents and logarithms. These concepts are fundamental to mathematics and are used extensively in various fields. By understanding these concepts, we can solve a wide range of problems involving exponents and logarithms.

References

  • [1] "Exponents and Logarithms" by Math Open Reference
  • [2] "Exponents and Logarithms" by Khan Academy
  • [3] "Exponents and Logarithms" by Wolfram MathWorld

Glossary

  • Exponent: A number that is raised to a power.
  • Logarithm: The inverse of an exponent.
  • Negative Exponent: A number that is raised to a negative power.
  • Product of Powers: A rule that states when we multiply two powers with the same base, we can add the exponents.
  • Power of a Power: A rule that states when we raise a power to another power, we can multiply the exponents.
    Exponents and Logarithms Q&A =============================

Introduction

Exponents and logarithms are fundamental concepts in mathematics that play a crucial role in various fields. In this article, we will answer some of the most frequently asked questions about exponents and logarithms.

Q: What is an exponent?

A: An exponent is a number that is raised to a power. For example, in the expression $2^3$, the exponent is $3$.

Q: What is a logarithm?

A: A logarithm is the inverse of an exponent. It is used to solve equations involving exponents. For example, $\log_2 8 = 3$, since $2^3 = 8$.

Q: What is a negative exponent?

A: A negative exponent is a number that is raised to a negative power. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: How do I evaluate a negative exponent?

A: To evaluate a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the product of powers rule?

A: The product of powers rule states that when you multiply two powers with the same base, you can add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: What is the power of a power rule?

A: The power of a power rule states that when you raise a power to another power, you can multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents and logarithms. For example, to solve the equation $2^x = 8$, you can take the logarithm of both sides and use the property of logarithms that states $\log_a (x \times y) = \log_a x + \log_a y$.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the inverse of an exponent. While an exponent tells you how many times to multiply a base by itself to get a certain value, a logarithm tells you what power the base must be raised to in order to get a certain value.

Q: How do I use logarithms to solve equations?

A: To use logarithms to solve equations, you can take the logarithm of both sides of the equation and use the properties of logarithms to simplify the equation. For example, to solve the equation $2^x = 8$, you can take the logarithm of both sides and use the property of logarithms that states $\log_a (x \times y) = \log_a x + \log_a y$.

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

  • log⁑a(xΓ—y)=log⁑ax+log⁑ay\log_a (x \times y) = \log_a x + \log_a y

  • (log⁑ax)n=log⁑a(xn)(\log_a x)^n = \log_a (x^n)

  • log⁑axy=log⁑axβˆ’log⁑ay\log_a \frac{x}{y} = \log_a x - \log_a y

Conclusion

In this article, we have answered some of the most frequently asked questions about exponents and logarithms. We hope that this article has been helpful in clarifying some of the concepts and rules related to exponents and logarithms.

References

  • [1] "Exponents and Logarithms" by Math Open Reference
  • [2] "Exponents and Logarithms" by Khan Academy
  • [3] "Exponents and Logarithms" by Wolfram MathWorld

Glossary

  • Exponent: A number that is raised to a power.
  • Logarithm: The inverse of an exponent.
  • Negative Exponent: A number that is raised to a negative power.
  • Product of Powers: A rule that states when you multiply two powers with the same base, you can add the exponents.
  • Power of a Power: A rule that states when you raise a power to another power, you can multiply the exponents.