Simplify log of 2x+ log of x- log of 2
The question provided is asking you to perform algebraic manipulations of logarithmic expressions. Specifically, it involves using log rules, such as the product rule, which states that the log of a product is the sum of the logs (log(a) + log(b) = log(ab)), and the quotient rule, which says that the log of a quotient is the difference of the logs (log(a) - log(b) = log(a/b)). The question wants you to combine the separate logarithmic terms, log of (2x), log of (x), and log of (2), into a single logarithmic expression.
$log \left(\right. 2 x \left.\right) + log \left(\right. x \left.\right) - log \left(\right. 2 \left.\right)$
Answer
Solution:
Step 1:
Combine the logs that are being added using the product rule for logarithms, which states $\log_b(x) + \log_b(y) = \log_b(xy)$. This gives us $\log(2x \cdot x) - \log(2)$.
Step 2:
Apply the quotient rule for logarithms, which says $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$. We get $\log\left(\frac{2x \cdot x}{2}\right)$.
Step 3:
Simplify the expression by combining like terms.
Step 3.1:
Group the $x$ terms together to get $\log\left(\frac{2(x \cdot x)}{2}\right)$.
Step 3.2:
Simplify the multiplication of $x$ with itself, which is $x^2$, resulting in $\log\left(\frac{2x^2}{2}\right)$.
Step 4:
Reduce the fraction by eliminating common factors.
Step 4.1:
Remove the common factor of $2$ to obtain $\log\left(\frac{\cancel{2}x^2}{\cancel{2}}\right)$.
Step 4.2:
Simplify the fraction further to just $x^2$, yielding the final result $\log(x^2)$.
Knowledge Notes:
The problem-solving process involves simplifying a logarithmic expression using logarithm properties. The relevant knowledge points include:
Product Property of Logarithms: This property states that the logarithm of a product is equal to the sum of the logarithms of the factors: $\log_b(xy) = \log_b(x) + \log_b(y)$.
Quotient Property of Logarithms: This property states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$.
Combining Like Terms: When simplifying expressions, we can add or subtract like terms, which are terms that have the same variable raised to the same power.
Simplifying Exponents: When multiplying like bases, we add the exponents: $x^m \cdot x^n = x^{m+n}$.
Reducing Fractions: A fraction can be simplified by dividing both the numerator and the denominator by their greatest common factor.
Using these properties and knowledge points, the original logarithmic expression can be simplified to a single logarithm.
