#1
1. Solve the exponential equation 2^x = 8.
x = 3
ExplanationExponent that gives 8 when applied to 2 is 3.
#2
2. Evaluate log base 5 of 25.
2
ExplanationLog base 5 of 25 gives the power that 5 must be raised to obtain 25, which is 2.
#3
11. Evaluate log base 4 of 1.
0
ExplanationLog base 4 of 1 gives the exponent to which 4 must be raised to obtain 1, which is 0.
#4
15. Simplify the expression log base 7 of (49/7).
1
Explanation49/7 = 7, log base 7 of 7 equals 1.
#5
21. Evaluate the expression ln(e^4).
4
Explanationln and e are inverse functions, so ln(e^4) simply equals 4.
#6
3. Solve the logarithmic equation log base 2 of (x + 1) = 3.
x = 7
ExplanationThe antilog of 3 base 2 is x + 1, which gives x = 7.
#7
4. If y = 3^(2x), what is the value of x when y = 81?
3
ExplanationSubstitute y with 81, solving for x gives x = 3.
#8
6. Simplify the expression log base 3 of 27 - log base 3 of 9.
2
ExplanationLog base 3 of 27 is 3, log base 3 of 9 is 2, subtracting gives 1.
#9
9. If y = e^(2x), what is the value of x when y = 10?
ln(2)
ExplanationSubstituting y with 10, solving for x gives x = ln(2).
#10
12. If y = 6^(x+2), what is the value of x when y = 216?
3
ExplanationSubstitute y with 216, solving for x gives x = 3.
#11
5. Solve the equation 4^x - 2(2^x) - 8 = 0.
x = 2
ExplanationBy substitution, the equation simplifies to (2^x - 4)(2^x + 2) = 0, giving x = 2.
#12
7. Solve the exponential equation 5^(2x+1) = 125.
x = 3
ExplanationRewriting 125 as 5^3, the equation simplifies to 2x + 1 = 3, giving x = 3.
#13
8. Find the value of x in the equation log base 2 of (x^2) = 4.
x = 4
ExplanationThe antilog of 4 base 2 is x^2, giving x = 4.
#14
10. Solve the logarithmic equation ln(x) - ln(x - 3) = 1.
x = 4
ExplanationApplying log rules, the equation simplifies to x = 4.
#15
14. Find the value of x in the equation e^(2x) - 3e^x + 2 = 0.
ln(2)
ExplanationSubstitute e^x with y, the equation becomes quadratic in y. Solving gives y = e^x = 2, hence x = ln(2).