if 3^a=5√3^4,what is the value of a?

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Identify Given Equation and Goal: Identify the given equation and the goal. We are given that 3^a = 5√(3^4), and we need to find the value of a.Rewrite Square Root as Exponent: Rewrite the square root of 3^4 as an exponent. The square root of a number is the same as raising that number to the power of 1/2. So, √(3^4) is the same as (3^4)^(1/2).Apply Power of Power Rule: Apply the power of a power rule. According to the power of a power rule, (x^m)^n = x^(m*n). Therefore, (3^4)^(1/2) = 3^(4*(1/2)).Simplify the Exponent: Simplify the exponent. 3^(4*(1/2)) simplifies to 3^2 because 4*(1/2) equals 2.Substitute Simplified Exponent: Substitute the simplified exponent back into the equation. Now we have 3^a = 5 * 3^2.Divide to Isolate 3^a: Divide both sides of the equation by 3^2 to isolate 3^a. Doing this, we get (3^a) / (3^2) = 5.Apply Quotient of Powers Rule: Apply the quotient of powers rule. According to the quotient of powers rule, a^m / a^n = a^(m-n). Therefore, (3^a) / (3^2) = 3^(a-2).Set Expression Equal to 5: Set the expression equal to 5. Now we have 3^(a-2) = 5.Solve for a with Logarithm: Solve for a by taking the logarithm of both sides. To solve for a, we can take the logarithm with base 3 of both sides, which gives us a - 2 = log_3(5).Isolate a by Adding 2: Isolate a by adding 2 to both sides. Adding 2 to both sides gives us a = log_3(5) + 2.Calculate log_3(5): Calculate the value of log_3(5). Using a calculator or logarithm properties, we find that log_3(5) is approximately 1.465 (rounded to three decimal places).Add 2 to log_3(5): Add 2 to the approximate value of log_3(5). Adding 2 to 1.465 gives us a ≈ 1.465 + 2.Calculate Final Approximate Value: Calculate the final approximate value of a. a ≈ 1.465 + 2 ≈ 3.465.

if $3^a=5\sqrt{3^4}$ ,what is the value of $a$ ?

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if $3^a=5\sqrt{3^4}$ ,what is the value of $a$ ?