Problem 6
Question
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$3^{2 x+1}=27$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = 1\).
1Step 1: Express both sides as powers of the same base
The given equation is \(3^{2 x+1}=27\). Since 27 can be written as \(3^3\), the equation can be rewritten as \(3^{2x+1} = 3^3\).
2Step 2: Equate Exponents
Since the bases on both sides of the equation are same (base 3), the exponents can be equated. Therefore, the equation becomes: \(2x + 1 = 3\).
3Step 3: Solve for x
Finally, solve the equation for x. Subtract 1 from both sides of the equation to get \(2x = 3 - 1 = 2\). Then divide both sides by 2 to isolate x: \(x = 2/2 = 1\).
Other exercises in this chapter
Problem 5
Write each equation in its equivalent exponential form. $$5=\log _{b} 32$$
View solution Problem 5
Approximate each number using a calculator. Round your answer to three decimal places. $$4^{-1.5}$$
View solution Problem 6
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
View solution Problem 6
Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$
View solution