#1
What is the solution to the exponential equation 2^x = 16?
4
ExplanationExponentiation: 2 raised to what power equals 16? Answer: 4.
#2
Solve the logarithmic equation log₃(81) = x.
3
ExplanationLogarithm base 3 of 81 equals x. x = 3.
#3
What is the base of the natural logarithm (ln)?
e
ExplanationBase of natural logarithm, denoted as 'e'.
#4
Solve the exponential equation 4^(x-1) = 16.
3
ExplanationExponential equation solution: 4 raised to what power minus 1 equals 16? Answer: 3.
#5
Solve the logarithmic equation log₂(x) + log₂(x + 4) = 3.
4
ExplanationLogarithmic equation: Log base 2 of x times log base 2 of x plus 4 equals 3. Solution: x = 4.
#6
What is the value of log₄(√16)?
1/2
ExplanationLogarithm base 4 of square root of 16 equals 1/2.
#7
Solve the logarithmic equation log₅(2x + 1) + log₅(x) = 2.
3
ExplanationLogarithmic equation: Log base 5 of 2x plus 1 plus log base 5 of x equals 2. Solution: x = 3.
#8
If log₅(x) = 2, what is the value of x?
25
ExplanationLogarithm base 5 of x equals 2. x = 25.
#9
Solve the exponential equation 3^(2x) = 27.
3
ExplanationExponential equation: 3 raised to what power times 2 equals 27? Answer: 3.
#10
Evaluate the logarithmic expression log₇(49) + log₇(7) - log₇(343).
0
ExplanationLogarithmic expression simplification: log base 7 of 49 plus log base 7 of 7 minus log base 7 of 343 equals 0.
#11
If e^(2x) = 16, what is the value of x?
2
ExplanationExponential equation: 'e' raised to what power times 2 equals 16? Answer: 2.
#12
If ln(y) = 3, what is the value of y?
e³
ExplanationNatural logarithm: ln(y) equals 3. Therefore, y = e raised to the power of 3.