Simplify/Condense log of 3x- log of 10
In the given problem, you are asked to combine or simplify the expression containing two logarithmic terms with the same base. Specifically, the expression involves the logarithm of a variable product (3x) from which the logarithm of a constant (10) is being subtracted. The task is to use the properties of logarithms to condense these two terms into a single logarithmic expression.
$log \left(\right. 3 x \left.\right) - log \left(\right. 10 \left.\right)$
Answer
Solution:
Step 1:
Apply the quotient rule for logarithms: $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$. Thus, we get $\log\left(\frac{3x}{10}\right)$.
Step 2:
Express the fraction $\frac{3x}{10}$ as a product involving the reciprocal of 10: $\log(3x \cdot 10^{-1})$.
Step 3:
Separate the logarithm of the product into the sum of logarithms: $\log(3x) + \log(10^{-1})$.
Step 4:
Utilize the power rule for logarithms to bring the exponent in front of the log: $\log(3x) - \log(10)$.
Step 5:
Recognize that the logarithm of 10 to base 10 is 1: $\log(3x) - 1 \cdot 1$.
Step 6:
Simplify the expression by multiplying -1 by 1: $\log(3x) - 1$.
Knowledge Notes:
The problem involves simplifying a logarithmic expression using logarithm properties. Here are the relevant knowledge points:
Quotient Rule: The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$.
Product Rule: The product rule for logarithms states that the logarithm of a product is the sum of the logarithms: $\log_b(xy) = \log_b(x) + \log_b(y)$.
Power Rule: The power rule for logarithms states that the logarithm of a power is the exponent times the logarithm of the base: $\log_b(x^y) = y \cdot \log_b(x)$.
Logarithm of Base 10: The logarithm of 10 to base 10 is 1 because $10^1 = 10$: $\log_{10}(10) = 1$.
Simplifying Expressions: Simplifying expressions involves applying mathematical rules and properties to rewrite expressions in a simpler or more standard form.
In the given problem, we use these properties to simplify the logarithmic expression from a difference of logarithms to a single logarithmic term minus a constant.
