Solve For \[$ P \$\] In The Equation:$\[ \frac{1}{3} \cdot \log _5 64 = \log _5 8 + \log _5 P \\]

Introduction

In this article, we will delve into the world of logarithms and explore a mathematical equation that involves logarithmic functions. The equation in question is $\frac{1}{3} \cdot \log _5 64 = \log _5 8 + \log _5 p$. Our goal is to solve for the variable $p$ in this equation. We will use various mathematical techniques and properties of logarithms to simplify the equation and isolate the variable $p$.

Understanding Logarithmic Functions

Before we dive into the solution, let's take a moment to understand the concept of logarithmic functions. A logarithmic function is the inverse of an exponential function. In other words, if $y = a^x$, then $x = \log_a y$. The logarithmic function $\log_a x$ gives us the exponent to which we must raise the base $a$ to obtain the number $x$.

Properties of Logarithms

There are several properties of logarithms that we will use to simplify the equation. These properties include:

• Product Property: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Property: $\log_a \frac{x}{y} = \log_a x - \log_a y$
• Power Property: $\log_a x^y = y \log_a x$

Simplifying the Equation

Now that we have a good understanding of logarithmic functions and their properties, let's simplify the equation. We can start by using the product property to expand the left-hand side of the equation:

$$\frac{1}{3} \cdot \log _5 64 = \log _5 (64^{\frac{1}{3}})$$

Using the power property, we can rewrite $64^{\frac{1}{3}}$ as $(2^6)^{\frac{1}{3}}$, which simplifies to $2^2$.

$$\log _5 (64^{\frac{1}{3}}) = \log _5 (2^2)$$

Now, we can use the power property again to rewrite $\log _5 (2^2)$ as $2 \log _5 2$.

$$2 \log _5 2 = \log _5 8 + \log _5 p$$

Isolating the Variable p

Now that we have simplified the equation, let's isolate the variable $p$. We can start by subtracting $\log _5 8$ from both sides of the equation:

$$2 \log _5 2 - \log _5 8 = \log _5 p$$

Using the quotient property, we can rewrite $2 \log _5 2 - \log _5 8$ as $\log _5 \frac{2^2}{8}$.

$$\log _5 \frac{2^2}{8} = \log _5 p$$

Now, we can use the quotient property again to rewrite $\log _5 \frac{2^2}{8}$ as $\log _5 \frac{4}{8}$.

$$\log _5 \frac{4}{8} = \log _5 p$$

Solving for p

Now that we have isolated the variable $p$, let's solve for $p$. We can start by rewriting $\log _5 \frac{4}{8}$ as $\log _5 \frac{1}{2}$.

$$\log _5 \frac{1}{2} = \log _5 p$$

Using the definition of logarithms, we can rewrite $\log _5 \frac{1}{2}$ as the exponent to which we must raise the base $5$ to obtain $\frac{1}{2}$.

$$5^{\log _5 \frac{1}{2}} = p$$

Using the property of logarithms that states $a^{\log_a x} = x$, we can simplify the left-hand side of the equation to $\frac{1}{2}$.

$$\frac{1}{2} = p$$

Conclusion

In this article, we have solved for the variable $p$ in the equation $\frac{1}{3} \cdot \log _5 64 = \log _5 8 + \log _5 p$. We used various mathematical techniques and properties of logarithms to simplify the equation and isolate the variable $p$. The final solution is $p = \frac{1}{2}$.

Final Answer

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Introduction

In our previous article, we solved for the variable $p$ in the equation $\frac{1}{3} \cdot \log _5 64 = \log _5 8 + \log _5 p$. We used various mathematical techniques and properties of logarithms to simplify the equation and isolate the variable $p$. In this article, we will answer some frequently asked questions related to the solution.

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm in the equation is 5.

Q: What is the value of $p$ in the equation?

A: The value of $p$ in the equation is $\frac{1}{2}$.

Q: How did you simplify the equation?

A: We simplified the equation by using the product property, power property, and quotient property of logarithms.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_a (xy) = \log_a x + \log_a y$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_a x^y = y \log_a x$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_a \frac{x}{y} = \log_a x - \log_a y$.

Q: How did you isolate the variable $p$?

A: We isolated the variable $p$ by subtracting $\log _5 8$ from both sides of the equation and then using the quotient property to rewrite the left-hand side of the equation.

Q: What is the final solution to the equation?

A: The final solution to the equation is $p = \frac{1}{2}$.

Q: Can you explain the concept of logarithmic functions?

A: A logarithmic function is the inverse of an exponential function. In other words, if $y = a^x$, then $x = \log_a y$. The logarithmic function $\log_a x$ gives us the exponent to which we must raise the base $a$ to obtain the number $x$.

Q: What are some common properties of logarithms?

A: Some common properties of logarithms include the product property, power property, and quotient property.

Conclusion

In this article, we have answered some frequently asked questions related to solving for the variable $p$ in the equation $\frac{1}{3} \cdot \log _5 64 = \log _5 8 + \log _5 p$. We have also reviewed the concept of logarithmic functions and some common properties of logarithms.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.