91Ó°ÊÓ

Logarithms have several useful properties that help simplify complex equations. One essential property is the subtraction rule: \(\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)\). This property allows us to combine logarithms into a single term when subtracting two logarithms with the same base. This is particularly useful when dealing with equations that involve multiple logarithmic terms.

In the original exercise, we applied this property to combine \(2 \log x\) and \(\log 10x\) into a single logarithmic function. By doing this, we could rewrite the expression as \(\log\left(\frac{x^2}{10x}\right)\), which simplifies further to \(\log\left(\frac{x}{10}\right)\) when we simplify the fraction. Remember:

• Combining logs can reduce complexity.
• Always ensure the bases are the same before applying the property.
• Simplifying fractions inside the log makes it easier to solve.

Converting a logarithmic equation to its exponential form is a key step in solving logarithmic equations. The fundamental idea behind logarithms is that they are the inverse of exponents. This relationship is captured in the property \(\log_a b = c\), which can be rewritten as \(a^c = b\).

In the problem, when we reach the equation \(\log\left(\frac{x}{10}\right) = 0\), we can convert it into an exponential form. Since the base of the logarithm is 10, this conversion results in the equation \(10^0 = \frac{x}{10}\). The property that any number raised to the power of zero equals one, \(a^0 = 1\), is crucial here.

• Exponential form allows a switch from a log equation to a simpler, numeric equation.
• Understanding the base and the exponent is key.
• This method helps in isolating variables for solving.

Once the logarithmic equation is transformed into its exponential form, the next step is simplifying and solving it. Remember, the goal is to isolate the variable to find its value. In this exercise, after converting to exponential form, we get \(10^0 = 1 = \frac{x}{10}\).

To solve for \(x\), we multiply both sides by 10 to get rid of the fraction, leading to \(x = 10\). This step demonstrates the process of isolating the variable and simplifying the equation as much as possible.

• Always aim to simplify equations to isolate the variable.
• Use inverse operations to eliminate fractions or coefficients.
• Check your solution by substituting back into the original equation.