Question

Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution.

11^x=137
The solution set expressed in terms of logarithms is
(Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use In for natural logarithm and log for common logarithm.)
Now use a calculator to obtain a decimal approximation for the solution
The solution set is
(Use a comma to separate answers as needed Round to two decimal places as needed)

Answer 1

The exponential equation 11^x = 137 can be solved using natural logarithms, resulting in the solution set x = ln(137)/ln(11). A calculator gives the decimal approximation x ≈ 2.05.

Step-by-step explanation:

To solve the exponential equation 11^x = 137, we need to use logarithms. Logarithms allow us to rewrite the equation in terms of exponents that can be solved for x. Choose either natural logarithms (ln) or common logarithms (log), depending on what is more convenient or if a specific type is requested.

Using natural logarithms, the equation would be:

1. Take the natural logarithm of both sides of the equation to get ln(11^x) = ln(137).
2. By the properties of logarithms, rewrite ln(11^x) as x * ln(11) = ln(137).
3. Divide both sides by ln(11) to isolate x, giving us x = ln(137)/ln(11).

Using a calculator to find the decimal approximation:

1. Calculate the natural logarithm of 137 to get approximately 4.91998.
2. Calculate the natural logarithm of 11 to get approximately 2.397895.
3. Divide the natural logarithm of 137 by the natural logarithm of 11 to find x ≈ 2.05 (rounded to two decimal places).

Thus, the solution set in terms of natural logarithms is x = ln(137)/ln(11) and the decimal approximation is x ≈ 2.05.