Using the Properties of Logarithms

Oct 13, 2025 by miodeya.com 10 views

Understanding the Problem

To evaluate $$\log_{12} y^2$$, we are given the value of $$\log_{12} y$$, which is equal to 16. This means that we need to find the value of $$\log_{12} y^2$$ using the given information.

Using the Properties of Logarithms

One of the properties of logarithms states that $$\log_a b^c = c \log_a b$$. This property allows us to rewrite the expression $$\log_{12} y^2$$ as $$2 \log_{12} y$$.

Applying the Given Information

We are given that $$\log_{12} y = 16$$. Using this information, we can substitute the value of $$\log_{12} y$$ into the expression $$2 \log_{12} y$$ to get $$2 \times 16 = 32$$.

Evaluating the Expression

Therefore, the value of $$\log_{12} y^2$$ is equal to 32.

Conclusion

In conclusion, we have evaluated the expression $$\log_{12} y^2$$ using the given information and the properties of logarithms. The final answer is 32.

Final Answer

The final answer is $\boxed{32}$ .

Step-by-Step Solution

Here's a step-by-step solution to the problem:

We are given that $$\log_{12} y = 16$$.
We need to find the value of $$\log_{12} y^2$$ using the given information.
We can use the property of logarithms that states $$\log_a b^c = c \log_a b$$ to rewrite the expression $$\log_{12} y^2$$ as $$2 \log_{12} y$$.
We can substitute the value of $$\log_{12} y$$ into the expression $$2 \log_{12} y$$ to get $$2 \times 16 = 32$$.
Therefore, the value of $$\log_{12} y^2$$ is equal to 32.

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

Not using the property of logarithms that states $$\log_a b^c = c \log_a b$$.
Not substituting the value of $$\log_{12} y$$ into the expression $$2 \log_{12} y$$.
Not simplifying the expression $$2 \log_{12} y$$ to get $$2 \times 16 = 32$$.

Real-World Applications

This problem has real-world applications in various fields such as engineering, physics, and computer science. For example, in engineering, logarithmic functions are used to model population growth, chemical reactions, and electrical circuits. In physics, logarithmic functions are used to model the behavior of particles in a gas, the decay of radioactive materials, and the behavior of electrical circuits. In computer science, logarithmic functions are used to model the time complexity of algorithms, the space complexity of data structures, and the performance of computer networks.

Conclusion

In conclusion, we have evaluated the expression $$\log_{12} y^2$$ using the given information and the properties of logarithms. The final answer is 32. This problem has real-world applications in various fields such as engineering, physics, and computer science.

Introduction

In the previous article, we evaluated the expression $$\log_{12} y^2$$ using the given information and the properties of logarithms. In this article, we will answer some frequently asked questions related to logarithmic expressions.

Q1: What is the property of logarithms that states $$\log_a b^c = c \log_a b$$?

A1: The property of logarithms that states $$\log_a b^c = c \log_a b$$ is known as the power rule of logarithms. This property allows us to rewrite the expression $$\log_a b^c$$ as $$c \log_a b$$.

Q2: How do I apply the power rule of logarithms to evaluate an expression?

A2: To apply the power rule of logarithms, you need to identify the base and the exponent in the expression. Then, you can rewrite the expression using the power rule of logarithms. For example, if you have the expression $$\log_2 x^3$$, you can rewrite it as $$3 \log_2 x$$.

Q3: What is the difference between $$\log_a b$$ and $$\log_b a$$?

A3: The expressions $$\log_a b$$ and $$\log_b a$$ are known as logarithmic functions with different bases. The expression $$\log_a b$$ represents the exponent to which the base $$a$$ must be raised to obtain the number $$b$$. On the other hand, the expression $$\log_b a$$ represents the exponent to which the base $$b$$ must be raised to obtain the number $$a$$.

Q4: How do I evaluate an expression with a negative exponent?

A4: To evaluate an expression with a negative exponent, you need to use the property of logarithms that states $$\log_a b^{-c} = -c \log_a b$$. For example, if you have the expression $$\log_2 x^{-3}$$, you can rewrite it as $$-3 \log_2 x$$.

Q5: What is the relationship between logarithmic and exponential functions?

A5: The logarithmic and exponential functions are inverse functions of each other. This means that if you have a logarithmic function $$\log_a x$$, you can rewrite it as an exponential function $$a^x$$. Similarly, if you have an exponential function $$a^x$$, you can rewrite it as a logarithmic function $$\log_a x$$.

Q6: How do I evaluate an expression with a logarithm of a logarithm?

A6: To evaluate an expression with a logarithm of a logarithm, you need to use the property of logarithms that states $$\log_a (\log_b x) = \frac{\log_c x}{\log_c a}$$, where $$c$$ is any positive real number. For example, if you have the expression $$\log_2 (\log_3 x)$$, you can rewrite it as $$\frac{\log_5 x}{\log_5 2}$$.

Q7: What is the difference between a logarithmic function and an exponential function?

A7: A logarithmic function represents the exponent to which a base must be raised to obtain a given number. On the other hand, an exponential function represents the result of raising a base to a given exponent.

Q8: How do I graph a logarithmic function?

A8: To graph a logarithmic function, you need to use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q9: What are some real-world applications of logarithmic functions?

A9: Logarithmic functions have many real-world applications in fields such as engineering, physics, and computer science. For example, logarithmic functions are used to model population growth, chemical reactions, and electrical circuits.

Q10: How do I solve a logarithmic equation?

A10: To solve a logarithmic equation, you need to isolate the logarithmic expression on one side of the equation. Then, you can use the properties of logarithms to simplify the expression and solve for the variable.

Conclusion

In conclusion, we have answered some frequently asked questions related to logarithmic expressions. We hope that this article has provided you with a better understanding of logarithmic functions and their applications.