Evaluate The Expression: $\log_{1.5} 1.5^{21}$.

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Introduction

In mathematics, logarithms are a fundamental concept that helps us solve equations and express complex relationships in a simpler form. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this article, we will evaluate the expression $\log_{1.5} 1.5^{21}$, which involves understanding the properties of logarithms and how to simplify expressions involving exponents and logarithms.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. This can be represented mathematically as:

$$\log_b x = y \iff b^y = x$$

For example, if we have $\log_{10} 100 = 2$, then $10^2 = 100$. This shows that the logarithm of 100 to the base 10 is 2, because $10^2$ equals 100.

Evaluating the Expression

Now, let's evaluate the expression $\log_{1.5} 1.5^{21}$. To do this, we need to understand the properties of logarithms and how to simplify expressions involving exponents and logarithms.

One of the key properties of logarithms is the power rule, which states that:

$$\log_b (x^y) = y \log_b x$$

Using this property, we can rewrite the expression $\log_{1.5} 1.5^{21}$ as:

$$\log_{1.5} 1.5^{21} = 21 \log_{1.5} 1.5$$

Simplifying the Expression

Now, we need to simplify the expression $21 \log_{1.5} 1.5$. To do this, we can use the fact that $\log_b b = 1$, because $b^1 = b$. Therefore, we can rewrite the expression as:

$$21 \log_{1.5} 1.5 = 21 \cdot 1 = 21$$

Conclusion

In conclusion, the expression $\log_{1.5} 1.5^{21}$ can be evaluated using the properties of logarithms. By applying the power rule and simplifying the expression, we find that the value of the expression is 21.

Final Answer

The final answer to the expression $\log_{1.5} 1.5^{21}$ is $\boxed{21}$.

Related Topics

• Logarithmic functions
• Exponential functions
• Properties of logarithms
• Power rule of logarithms

Example Problems

• Evaluate the expression $\log_{2} 2^{15}$.
• Simplify the expression $\log_{3} (3^4 \cdot 3^2)$.
• Evaluate the expression $\log_{4} 4^{12}$.

Further Reading

• Logarithmic functions: A comprehensive guide
• Exponential functions: A comprehensive guide
• Properties of logarithms: A comprehensive guide
• Power rule of logarithms: A comprehensive guide
  

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Introduction

In our previous article, we evaluated the expression $\log_{1.5} 1.5^{21}$ using the properties of logarithms. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the value of $\log_{1.5} 1.5^{21}$?

A: The value of $\log_{1.5} 1.5^{21}$ is 21.

Q: How do you evaluate the expression $\log_{1.5} 1.5^{21}$?

A: To evaluate the expression $\log_{1.5} 1.5^{21}$, you can use the power rule of logarithms, which states that $\log_b (x^y) = y \log_b x$. This allows you to rewrite the expression as $21 \log_{1.5} 1.5$, which simplifies to 21.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_b (x^y) = y \log_b x$. This means that when you have a logarithm of a power, you can rewrite it as the exponent times the logarithm of the base.

Q: How do you simplify the expression $\log_{1.5} 1.5^{21}$?

A: To simplify the expression $\log_{1.5} 1.5^{21}$, you can use the fact that $\log_b b = 1$, because $b^1 = b$. This allows you to rewrite the expression as $21 \log_{1.5} 1.5$, which simplifies to 21.

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. This means that if you have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$.

Q: How do you evaluate the expression $\log_{2} 2^{15}$?

A: To evaluate the expression $\log_{2} 2^{15}$, you can use the power rule of logarithms, which states that $\log_b (x^y) = y \log_b x$. This allows you to rewrite the expression as $15 \log_{2} 2$, which simplifies to 15.

Q: How do you simplify the expression $\log_{3} (3^4 \cdot 3^2)$?

A: To simplify the expression $\log_{3} (3^4 \cdot 3^2)$, you can use the product rule of logarithms, which states that $\log_b (x \cdot y) = \log_b x + \log_b y$. This allows you to rewrite the expression as $\log_{3} 3^4 + \log_{3} 3^2$, which simplifies to 6 + 2 = 8.

Conclusion

In conclusion, evaluating the expression $\log_{1.5} 1.5^{21}$ requires an understanding of the properties of logarithms, including the power rule and the fact that $\log_b b = 1$. By applying these properties, we can simplify the expression and find its value.

Final Answer

The final answer to the expression $\log_{1.5} 1.5^{21}$ is $\boxed{21}$.

Related Topics

• Logarithmic functions
• Exponential functions
• Properties of logarithms
• Power rule of logarithms
• Product rule of logarithms

Example Problems

• Evaluate the expression $\log_{2} 2^{15}$.
• Simplify the expression $\log_{3} (3^4 \cdot 3^2)$.
• Evaluate the expression $\log_{4} 4^{12}$.

Further Reading

• Logarithmic functions: A comprehensive guide
• Exponential functions: A comprehensive guide
• Properties of logarithms: A comprehensive guide
• Power rule of logarithms: A comprehensive guide
• Product rule of logarithms: A comprehensive guide