Solve the following exponential equation by taking the natur

Solve the following exponential equation by taking the natural logarithm on both sides. Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 5 e^5x = 1480 What is the solution in terms of natural logarithms? The solution set is {}. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression.) What is the decimal approximation for the solution? The solution set is {}. (Use a comma to separate answers as needed. Round to two decimal places as needed.) 6. Solve the following exponential equation by taking the natural logarithm on both sides. Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution. e^4-7x = 916 What is the solution in terms of natural logarithms? The solution set is {}. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression.) What is the decimal approximation for the solution? The solution set is {}. (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Solution

5. 5e^x = 1480 so that e^x = 1480/5 = 3296. On taking natural logarithms of both the sides, we have ln(e^x) = ln 296 or, x ln e = ln 296 or, x = ln 296 ( as log a^b = b log a and ln e = 1). Thus x = ln 296 . The solution set is x = { ln 296}

Since ln 296 = = 5.690359454, the decimal approximation for the solution is 5.69. The solution set is { 5.69}

6. e^4-7x = 916. On taking natural logarithms of both the sides, we have ln ( e^4-7x) = ln 916 or, (4-7x)ln e = ln 916 or, 4 - 7x = ln 916, or, 7x = 4 - ln 916.Thus x = (1/7)( 4 -ln 916). The solution set is { (1/7)( 4 -ln 916)}. Since ln 916 = 6.820016365, the decimal approximation for the solution is 1/7(4 - 6.82) or, 1/7( -2.82) or, 0.402857142 or, - 0.40.   The solution set is { - 0.40}