Express as a difference of logarithms.
[tex]$\log \frac{v}{s}$[/tex]
[tex]$\log \frac{v}{s}=\square$[/tex]
Answer (2)
Apply the quotient rule of logarithms.
Express the logarithm of the quotient as the difference of the logarithms of the numerator and the denominator.
Therefore, lo g s v = lo g v − lo g s .
The final answer is lo g v − lo g s .
Explanation
Understanding the Problem We are asked to express the logarithm of a quotient as a difference of logarithms. This involves using the properties of logarithms to rewrite the given expression in a different form.
Applying the Quotient Rule The quotient rule of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, this is expressed as: lo g b y x = lo g b x − lo g b y
Expressing as a Difference Applying this rule to the given expression, we have: lo g s v = lo g v − lo g s
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale or calculating the pH of a solution in chemistry. The ability to express a logarithm of a quotient as a difference of logarithms can simplify calculations in these areas. For example, if you need to find the logarithm of a ratio of two earthquake intensities, you can subtract the logarithms of the individual intensities instead of calculating the ratio first. This can be particularly useful when dealing with very large or very small numbers. Understanding logarithmic properties allows scientists and engineers to simplify complex calculations and make accurate predictions.
We can express lo g s v as a difference of logarithms using the quotient rule, resulting in lo g v − lo g s . This property simplifies calculations involving logarithms of fractions. Essentially, it states that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
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