$100^v$ Find The Value To Get An Integer.

Introduction

In this discussion, we will explore the concept of finding the value of $v$ in the expression $100^v$ such that the result is an integer. We will use the given information about the variable $r$ in the expression $64^r$ to help us solve for $v$. The given solution for $r$ is $\frac{1}{3}$, but we are also told that $\frac{5}{6}$ is the answer. Furthermore, if the variable $r$ is equal to $1 \frac{2}{3}$, it means that the result is an integer.

Understanding the Problem

To begin, let's analyze the problem and understand what is being asked. We are given the expression $100^v$ and we need to find the value of $v$ such that the result is an integer. This means that $100^v$ must be a whole number, without any fractional or decimal parts.

Using the Given Information

We are given that $64^r = \frac{1}{3}$ is one solution, but we are also told that $\frac{5}{6}$ is the answer. This means that we have two different values for $r$ that satisfy the equation. We can use this information to help us solve for $v$.

Solving for $v$

Q&A

Q: What is the problem asking for?

A: The problem is asking us to find the value of $v$ in the expression $100^v$ such that the result is an integer.

Q: What information is given about the variable $r$ in the expression $64^r$?

A: We are given that $64^r = \frac{1}{3}$ is one solution, but we are also told that $\frac{5}{6}$ is the answer. Furthermore, if the variable $r$ is equal to $1 \frac{2}{3}$, it means that the result is an integer.

Q: How can we use the given information to help us solve for $v$?

A: We can use the fact that $64^r = \frac{1}{3}$ to help us solve for $v$. By analyzing the relationship between $64^r$ and $100^v$, we can determine the value of $v$ that will result in an integer.

Q: What is the relationship between $64^r$ and $100^v$?

A: The relationship between $64^r$ and $100^v$ is that they are both exponential expressions. We can use the properties of exponents to help us solve for $v$.

Q: How can we use the properties of exponents to help us solve for $v$?

A: We can use the property of exponents that states $a^b = c \Rightarrow b = \log_a c$. We can apply this property to the expression $64^r = \frac{1}{3}$ to help us solve for $r$. Once we have the value of $r$, we can use it to find the value of $v$.

Q: What is the value of $r$ that satisfies the equation $64^r = \frac{1}{3}$?

A: To find the value of $r$, we can use the property of exponents that states $a^b = c \Rightarrow b = \log_a c$. Applying this property to the expression $64^r = \frac{1}{3}$, we get $r = \log_{64} \frac{1}{3}$.

Q: How can we simplify the expression $r = \log_{64} \frac{1}{3}$?

A: We can simplify the expression $r = \log_{64} \frac{1}{3}$ by using the change of base formula. The change of base formula states that $\log_a b = \frac{\log_c b}{\log_c a}$. Applying this formula to the expression $r = \log_{64} \frac{1}{3}$, we get $r = \frac{\log \frac{1}{3}}{\log 64}$.

Q: What is the value of $r$ that satisfies the equation $r = \frac{\log \frac{1}{3}}{\log 64}$?

A: To find the value of $r$, we can use a calculator to evaluate the expression $\frac{\log \frac{1}{3}}{\log 64}$. This will give us the value of $r$ that satisfies the equation.

Q: How can we use the value of $r$ to find the value of $v$?

A: Once we have the value of $r$, we can use it to find the value of $v$. We can do this by using the property of exponents that states $a^b = c \Rightarrow b = \log_a c$. Applying this property to the expression $100^v$, we get $v = \log_{100} 100^r$.

Q: What is the value of $v$ that satisfies the equation $v = \log_{100} 100^r$?

A: To find the value of $v$, we can use the property of exponents that states $a^b = c \Rightarrow b = \log_a c$. Applying this property to the expression $100^v$, we get $v = \log_{100} 100^r = r$.

Q: What is the value of $v$ that satisfies the equation $v = r$?

A: Since we have found that $r = \frac{\log \frac{1}{3}}{\log 64}$, we can substitute this value into the equation $v = r$ to get $v = \frac{\log \frac{1}{3}}{\log 64}$.

Q: What is the value of $v$ that satisfies the equation $v = \frac{\log \frac{1}{3}}{\log 64}$?

A: To find the value of $v$, we can use a calculator to evaluate the expression $\frac{\log \frac{1}{3}}{\log 64}$. This will give us the value of $v$ that satisfies the equation.

Q: What is the final answer to the problem?

A: The final answer to the problem is the value of $v$ that satisfies the equation $100^v = \text{integer}$. This value is $\boxed{\frac{\log \frac{1}{3}}{\log 64}}$.