Question

Use the properties of logarithmic functions to simplify the expression on the left side of the equation and determine the values of x and y. Then evaluate the simplified expression. The value of x is , and the value of y is. The value of the expression, rounded to nearest hundredth, is .

Answer 1

To simplify the equation using logarithmic functions, take the natural logarithm of both sides and use properties of logarithms to solve for x and y. Then substitute the values in the simplified expression to evaluate it.

Step-by-step explanation:

To simplify the expression on the left side of the equation using the properties of logarithmic functions, let's work step by step:

1. Take the natural logarithm (ln) of both sides of the equation. The natural logarithm cancels the exponential function.
2. The natural logarithm of 5.6/16.0 is -1.050.
3. Now, we have the equation ln(x) - ln(2y) = -1.050.
4. Using the property of logarithms, subtracting the logarithms of two numbers is equivalent to dividing the numbers. So we have ln(x/2y) = -1.050.
5. To find the value of x/2y, take the inverse natural logarithm (e^) of both sides.
6. So we have x/2y = e^(-1.050).
7. To solve for x, multiply both sides of the equation by 2y.
8. Therefore, x = 2y * e^(-1.050).
9. Once you have the values for x and y, substitute them into the simplified expression to find its value.

Answer 2

4x = 16log 4x = log 16 Take the common logarithm of both sides. (Remember, when no base is written, that means the base is 10.) What can you do with that new equation? log 4x = log 16x log 4 = log 16Use the power property of logarithms to simplify the logarithm on the left side of the equation. x log 4 = log 16Remember that log 4 is a number. You can divide both sides of the equation by log 4 to get x by itself.AnswerUse a calculator to evaluate the logarithms and the quotient.