Question

Use the rules of logarithms to find the value of x. Verify your answer with a calculator a log(x) = log(2) + log(6)

b. log(x) = log(24) - log(2) c. log(x ^ 2) = 2 * log(12) d . log(x) = 4 * log(2) - 3 * log(2)

Answer 1

To find the value of x using the rules of logarithms, we can simplify the equations using the properties of logarithms. The value of x for each equation is as follows: a. x = 12, b. x = 12, c. x = 12, d. x = 2.

Step-by-step explanation:

To find the value of x using the rules of logarithms, we need to use the properties of logarithms to simplify the equations.

a. log(x) = log(2) + log(6)

Using the property that the logarithm of the number resulting from the multiplication of two numbers is the sum of the logarithms of the two numbers, we can simplify the equation:

log(x) = log(2) + log(6) = log(2 * 6) = log(12)

Therefore, x = 12.

b. log(x) = log(24) - log(2)

Using the property that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, we can simplify the equation:

log(x) = log(24) - log(2) = log(24/2) = log(12)

Therefore, x = 12.

c. log(x ^ 2) = 2 * log(12)

Using the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we can simplify the equation:

log(x ^ 2) = 2 * log(12)

2 * log(x) = 2 * log(12)

log(x) = log(12)

Therefore, x = 12.

d. log(x) = 4 * log(2) - 3 * log(2)

Using the property that the logarithm of the number resulting from the multiplication of two numbers is the sum of the logarithms of the two numbers, we can simplify the equation:

log(x) = 4 * log(2) - 3 * log(2) = log(2^4) - log(2^3) = log(16) - log(8)

Using the property that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, we can further simplify the equation:

log(x) = log(16) - log(8) = log(16/8) = log(2)

Therefore, x = 2.

Answer 2

To solve the given logarithmic equations for x, we apply the properties of logarithms: product, quotient, and power rules. The solutions found are x = 12, x = 12, x = ±12, and x = 2 for the respective equations. These values can be verified with a calculator using inverse logarithm functions.

Step-by-step explanation:

Understanding Logarithms to Find the Value of x

To find the value of x using the rules of logarithms, we'll solve each equation one by one:

1. log(x) = log(2) + log(6): By the property of logarithms that states the sum of logarithms is the logarithm of the product of numbers, we get log(x) = log(12). That implies x = 12, because if the logs are equal, so are the numbers.
2. log(x) = log(24) - log(2): By the property that the logarithm of the division of two numbers is the difference between the logarithms of the two numbers, we get log(x) = log(12). Similarly, this implies x = 12.
3. log(x^2) = 2 * log(12): The property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number gives us log(x^2) = log(12^2), which means log(x^2) = log(144). Therefore, x^2 = 144, and x = ±12.
4. log(x) = 4 * log(2) - 3 * log(2): Simplifying this gives us log(x) = log(2). Consequently, x = 2.

After calculating these values, you can verify them using a calculator. To calculate a number from its logarithm, take the inverse log of the logarithm or calculate 10^x (where x is the logarithm of the number).