If $2 \log a+3 \log b=2$ then $a^2 b^3=$
Answer (2)
Use the logarithm property n lo g x = lo g x n to rewrite the given equation.
Use the logarithm property lo g x + lo g y = lo g ( x y ) to rewrite the equation as lo g ( a 2 b 3 ) = 2 .
Rewrite the equation in exponential form as a 2 b 3 = 1 0 2 .
Therefore, a 2 b 3 = 100 .
Explanation
Understanding the Problem We are given the equation 2 lo g a + 3 lo g b = 2 and we want to find the value of a 2 b 3 .
Applying Logarithm Properties Using the logarithm property n lo g x = lo g x n , we can rewrite the given equation as lo g a 2 + lo g b 3 = 2 .
Combining Logarithms Next, we use the logarithm property lo g x + lo g y = lo g ( x y ) to rewrite the equation as lo g ( a 2 b 3 ) = 2 .
Converting to Exponential Form Assuming the base of the logarithm is 10, we can rewrite the equation in exponential form as a 2 b 3 = 1 0 2 .
Finding the Value of a^2b^3 Therefore, a 2 b 3 = 100 .
Examples
Logarithms are incredibly useful in various fields, such as calculating the magnitude of earthquakes on the Richter scale or determining the pH levels of solutions in chemistry. In finance, they help in calculating returns on investments and analyzing growth rates. For instance, if you want to understand how quickly an investment will double at a certain interest rate, logarithms can provide a straightforward way to find the answer. They also play a crucial role in computer science for analyzing algorithm efficiency and data compression techniques.
The value of a 2 b 3 is 100 . This is derived by using properties of logarithms to rewrite the given equation and converting it into exponential form.
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