Solving Logarithmic Equations Finding The Value Of X

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of logarithms to solve a pretty interesting problem. We're tasked with finding the value of x in the equation ${ \log_8 x^3 - \log_x 8 = 2 }$, keeping in mind that x cannot be 8. Buckle up, because we're about to embark on a logarithmic journey!

Understanding the Problem

Before we jump into the solution, let's break down what we're dealing with. The equation involves logarithms with different bases, which can seem a bit daunting at first. But don't worry, we'll tackle it step by step.

Our main keywords here are logarithms, change of base, and solving equations. We need to manipulate the equation using logarithmic properties to isolate x. The condition ${ x \neq 8 }$ is crucial because it tells us that 8 is not a solution, and we need to be mindful of this constraint as we proceed.

Diving into Logarithmic Properties

Logarithmic properties are the key to unlocking this problem. Remember these fundamental rules?

1. ${ \log_b a^c = c \log_b a }$ (Power Rule)
2. ${ \log_b a = \frac{\log_k a}{\log_k b} }$ (Change of Base Rule)

These two properties are going to be our best friends in this mathematical quest. The power rule allows us to bring exponents outside the logarithm, simplifying expressions. The change of base rule is super handy when we have logarithms with different bases, as it allows us to express them in terms of a common base.

Cracking the Equation Step-by-Step

Let's start by applying the power rule to the first term in our equation:

${ \log_8 x^3 = 3 \log_8 x }$

Now our equation looks like this:

${ 3 \log_8 x - \log_x 8 = 2 }$

The next step is where the change of base rule comes into play. We can change the base of either logarithm, but it's often easiest to change the base of the second term to match the first. Let's change ${ \log_x 8 }$ to base 8:

${ \log_x 8 = \frac{\log_8 8}{\log_8 x} = \frac{1}{\log_8 x} }$

Remember that ${ \log_8 8 = 1 }$ because 8 raised to the power of 1 is 8. Now our equation transforms into:

${ 3 \log_8 x - \frac{1}{\log_8 x} = 2 }$

A Clever Substitution

To make things even cleaner, let's use a substitution. Let's say:

${ y = \log_8 x }$

Now our equation looks like a quadratic equation in disguise:

${ 3y - \frac{1}{y} = 2 }$

To get rid of the fraction, we can multiply the entire equation by y:

${ 3y^2 - 1 = 2y }$

Rearranging the terms, we get a standard quadratic equation:

${ 3y^2 - 2y - 1 = 0 }$

Solving the Quadratic Equation

Now, we need to solve this quadratic equation. We can do this by factoring, using the quadratic formula, or completing the square. Factoring is often the quickest method if it's possible. Let's see if we can factor this one.

We're looking for two numbers that multiply to ${ 3 \times -1 = -3 }$ and add up to -2. Those numbers are -3 and 1. So we can rewrite the middle term:

${ 3y^2 - 3y + y - 1 = 0 }$

Now, let's factor by grouping:

${ 3y(y - 1) + 1(y - 1) = 0 }$

${ (3y + 1)(y - 1) = 0 }$

Setting each factor equal to zero gives us two possible values for y:

${ 3y + 1 = 0 \Rightarrow y = -\frac{1}{3} }$

${ y - 1 = 0 \Rightarrow y = 1 }$

Back to x: Unraveling the Substitution

Remember, we're not looking for y; we're after x! We need to substitute back our original expression for y:

${ y = \log_8 x }$

So we have two cases to consider:

1. When ${ y = -\frac{1}{3} }$: ${ \log_8 x = -\frac{1}{3} }$ Converting to exponential form: ${ x = 8^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{8}} = \frac{1}{2} }$

2. When ${ y = 1 }$: ${ \log_8 x = 1 }$ Converting to exponential form: ${ x = 8^1 = 8 }$

The Final Verdict

We've found two possible values for x: ${ \frac{1}{2} }$ and 8. However, we have a condition that ${ x \neq 8 }$. So, we must discard the solution x = 8.

Therefore, the only valid solution is:

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

And that's it! We've successfully navigated the logarithmic landscape and found the value of x. Remember, the key to solving these problems is understanding and applying logarithmic properties, and a little bit of algebraic manipulation. Keep practicing, and you'll become a log-solving pro in no time!

Conclusion

In summary, finding the value of x in the equation ${ \log_8 x^3 - \log_x 8 = 2 }$ involves a series of steps, including applying logarithmic properties such as the power rule and change of base rule, making a substitution to simplify the equation, solving the resulting quadratic equation, and finally, substituting back to find x. The condition ${ x \neq 8 }$ is crucial in determining the final solution. The only valid solution we found is ${ x = \frac{1}{2} }$. This exercise highlights the importance of understanding and applying logarithmic properties to solve complex mathematical problems.

This was a fun one, wasn't it? Keep exploring the world of math, and you'll discover even more fascinating concepts and problems to solve. Until next time, happy problem-solving!