Find the solution of the exponential equation, correct to four decimal places.
4^(2x − 1) = 6
log ( a ^ n ) = n * log ( a )
( 2 x - 1 ) * log ( 4 ) = log ( 6 ) Divide both sides by log ( 4 )
2 x - 1 = log ( 6 ) / log ( 4 )
2 x = log ( 6 ) / log ( 4 ) + 1
2 x = log ( 6 ) / log ( 4 ) + log ( 4 ) / log ( 4 )
2 x = [ log ( 6 ) + log ( 4 ) ] / log 4
2 x = log ( 24 ) / log ( 4 ) Divide both sides by 2
x = [ log ( 24 ) / log ( 4 ) ] / 2
x = log ( 24 ) / 2 log ( 4 )
Remark:
log ( a ) + log ( b ) = log ( a * b )
log ( 4 ) + log ( 6 ) = log ( 6 * 4 ) = log ( 24 )
To find the solution of the exponential equation 4^(2x - 1) = 6, we need to isolate the variable x.
Step 1: Take the logarithm of both sides of the equation.
log(4^(2x - 1)) = log(6)
Step 2: Apply the logarithm property which states that log(a^b) = b * log(a).
(2x - 1) * log(4) = log(6)
Step 3: Simplify the equation by evaluating the logarithm of 4 and the logarithm of 6.
(2x - 1) * 0.6021 = 0.7782
Step 4: Divide both sides of the equation by 0.6021 to isolate the variable x.
2x - 1 = 0.7782 / 0.6021
Step 5: Simplify the equation.
2x - 1 = 1.2918
Step 6: Add 1 to both sides of the equation.
2x = 1.2918 + 1
Step 7: Simplify the equation.
2x = 2.2918
Step 8: Divide both sides of the equation by 2 to solve for x.
x = 2.2918 / 2
Step 9: Simplify the equation.
x ≈ 1.1459
Therefore, the solution to the exponential equation 4^(2x - 1) = 6, correct to four decimal places, is x ≈ 1.1459.