Solve For \[$ X \$\].$\[ \log X + \log 8 = 2 \\]

**Solve for $x$: $\log x + \log 8 = 2$**

Understanding the Problem

The given equation is a logarithmic equation, and our goal is to solve for the variable $x$. This equation involves the sum of two logarithmic terms, and we need to use the properties of logarithms to simplify and solve it.

What is a Logarithmic Equation?

A logarithmic equation is an equation that involves a logarithmic expression. In this case, the equation is $\log x + \log 8 = 2$. The logarithmic expression is $\log x$, which represents the power to which a base number (in this case, 10) must be raised to produce the number $x$.

Properties of Logarithms

To solve this equation, we need to use the properties of logarithms. One of the key properties is the product rule, which states that $\log a + \log b = \log (ab)$. This means that the sum of two logarithmic terms is equal to the logarithm of their product.

Solving the Equation

Using the product rule, we can rewrite the equation as:

$\log x + \log 8 = \log (x \cdot 8)$

This simplifies to:

$\log (8x) = 2$

What does this Equation Mean?

This equation means that the logarithm of $8x$ is equal to 2. In other words, $8x$ is equal to $10^2$, which is 100.

Solving for $x$

To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by dividing both sides of the equation by 8:

$\frac{\log (8x)}{1} = \frac{2}{1}$

This simplifies to:

$\log (8x) = 2$

Taking the exponential function of both sides, we get:

$8x = 10^2$

Dividing both sides by 8, we get:

$x = \frac{10^2}{8}$

Simplifying the Expression

To simplify the expression, we can use the fact that $10^2 = 100$. Therefore:

$x = \frac{100}{8}$

Final Answer

The final answer is:

$x = \frac{100}{8}$

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 4. Therefore:

$x = \frac{25}{2}$

Conclusion

In this article, we solved the logarithmic equation $\log x + \log 8 = 2$ using the properties of logarithms. We used the product rule to simplify the equation and then isolated $x$ on one side of the equation. The final answer is $x = \frac{25}{2}$.

Frequently Asked Questions

Q: What is a logarithmic equation? A: A logarithmic equation is an equation that involves a logarithmic expression.

Q: What is the product rule of logarithms? A: The product rule of logarithms states that $\log a + \log b = \log (ab)$.

Q: How do I solve a logarithmic equation? A: To solve a logarithmic equation, you need to use the properties of logarithms, such as the product rule, to simplify the equation and then isolate the variable on one side of the equation.

Q: What is the final answer to the equation $\log x + \log 8 = 2$? A: The final answer is $x = \frac{25}{2}$.

Q: How do I simplify a fraction? A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor.

Q: What is the greatest common divisor of 100 and 8? A: The greatest common divisor of 100 and 8 is 4.

Q: How do I divide a fraction by a number? A: To divide a fraction by a number, you need to multiply the fraction by the reciprocal of the number.